Matrices
Class 12th Mathematics RS Aggarwal Solution
- If a = [ {cccc} {5}&{-2}&{6}&{1} {7}&{0}&{8}&{-3} { root {2} } & { {3}/{5} }…
- Write the order of each of the following matrices:i. a = [ {cccc}…
- If a matrix has 18 elements, what are the possible orders it can have?…
- Find all possible orders of matrices having 7 elements.
- Construct a 3 × 2 matrix whose elements are given by aij = (2i – j).…
- Construct a 4 × 3 matrix whose elements are given by a_{ij} = {i}/{j}…
- Construct a 2 × 2 matrix whose elements are a_{ij} = { (i+2j)^{2} }/{2}…
- Construct a 2 × 3 matrix whose elements are a_{ij} = { (i-2j)^{2} }/{2}…
- Construct a 3 × 4 matrix whose elements are given by a_{ij} = {1}/{2} |-3i+j|…
- If a = [ {ccc} {2}&{-3}&{5} {-1}&{0}&{3} ] and b = [ {ccc} {3}&{2}&{-2}…
- If a = [ {cc} {3}&{5} {-2}&{0} {6}&{-1} ] , b = [ {cc} {-1}&{-3} {4}&{2}…
- If a = [ {ccc} {3}&{1}&{2} {1}&{2}&{-3} ] and b = [ {ccc} {-2}&{0}&{4}…
- Let a = [ {ll} {2}&{4} {3}&{2} ] , b = [ {cc} {1}&{3} {-2}&{5} ] and c = […
- Let a = [ {ccc} {0}&{1}&{-2} {5}&{-1}&{-4} ] , b = [ {ccc} {1}&{-3}&{-1}…
- If 5a = [ {ccc} {5}&{10}&{-15} {2}&{3}&{4} {1}&{0}&{-5} ] find A.…
- Find matrices A and B, if a+b = [ {ccc} {1}&{0}&{2} {5}&{4}&{-6} {7}&{3}&{8} ]…
- Find matrices A and B, if 2a-b = [ {rrr} {6}&{-6}&{0} {-4}&{2}&{1} ] and 2b+a…
- Find matrix X, if [ {ccc} {3}&{5}&{-9} {-1}&{4}&{-7} ] +x = [ {lll}…
- If a = [ {cc} {-2}&{3} {4}&{5} {1}&{-6} ] and b = [ {cc} {5}&{2} {-7}&{3}…
- Find the matrix X such that 2A – B + X = O,where a = [ {ll} {3}&{1} {0}&{2} ]…
- If a = [ {rrr} {1}&{-3}&{2} {2}&{0}&{2} ] and b = [ {ccc} {2}&{-1}&{-1}…
- If A = diag [2, -5, 9], B = diag [-3, 7, 14] and C = diag [4, -6, 3], find:(i)…
- Find the value of x and y, wheni. [ {x+y} {x-y} ] = [ {8} {4} ]
- Find the value of (x + y) from the following equation : 2 [ {cc} {1}&{3}…
- If [ {cc} {x-y}&{2y} {2y+z}&{x+y} ] = [ {cc} {1}&{4} {9}&{5} ] then write the…
- a = [ {cc} {2}&{-1} {3}&{0} {-1}&{4} ] and b = [ {rr} {-2}&{3} {0}&{4} ]…
- a = [ {ll} {-1}&{1} {-2}&{2} {-3}&{3} ] and b = [ {ccc} {3}&{-2}&{1}…
- a = [ {rrr} {0}&{1}&{-5} {2}&{4}&{0} ] and b = [ {cc} {1}&{3} {-1}&{0}…
- A = [1 2 3 4] and b = [ {1} {2} {3} {4} ] Compute AB and BA, which ever exists…
- a = [ {cc} {2}&{1} {3}&{2} {-1}&{1} ] and b = [ {rrr} {1}&{0}&{1}…
- a = [ {ll} {5}&{-1} {6}&{7} ] and b = [ {ll} {2}&{1} {3}&{4} ] Show that AB ≠…
- a = [ {lll} {1}&{2}&{3} {0}&{1}&{0} {1}&{1}&{0} ] and b = [ {ccc}…
- a = [ {cc} {costheta }&{-sintegrate heta} {sintheta}&{costheta} ] and b = […
- a = [ {lll} {1}&{2}&{1} {3}&{4}&{2} {1}&{3}&{2} ] and b = [ {ccc}…
- a = [ {lll} {1}&{3}&{-1} {2}&{2}&{-1} {3}&{0}&{-1} ] and b = [ {rrr}…
- If a = [ {ccc} {2}&{-3}&{-5} {-1}&{4}&{5} {1}&{-3}&{-4} ] and b = [ {ccc}…
- If a = [ {ccc} {0}&{c}&{-b} {-c}&{0}&{a} {b}&{-a}&{0} ] and b = [ {lll} {…
- a = [ {rrr} {2}&{3}&{-1} {3}&{0}&{2} ] , b = [ {1} {1} {2} ] and C = [1 -2]…
- a = [ {ll} {1}&{2} {3}&{4} ] , b = [ {cc} {2}&{0} {1}&{-3} ] and c = [ {cc}…
- a = [ {cc} {2}&{3} {-1}&{4} {0}&{1} ] , b = [ {cc} {5}&{-3} {2}&{1} ] and c =…
- If a = [ {ccc} {1}&{0}&{-2} {3}&{-1}&{0} {-2}&{1}&{1} ] , b = [ {ccc}…
- If a = [ {cc} {ab}& { b^{2} } { - a^{2} } &{-ab} ] show that A2 = O.…
- If a = [ {ccc} {2}&{-2}&{-4} {-1}&{3}&{4} {1}&{-2}&{-3} ] show that A2 = A.…
- If a = [ {ccc} {4}&{-1}&{-4} {3}&{0}&{-4} {3}&{-1}&{-3} ] show that A2 = I.…
- If a = [ {cc} {2}&{-1} {3}&{2} ] and b = [ {cc} {0}&{4} {-1}&{7} ] find…
- If a = [ {cc} {2}&{-2} {-3}&{4} ] then find (-A2 + 6A).
- If a = [ {cc} {3}&{1} {-1}&{2} ] show that (A2 – 5A + 7I) = O.
- Show that the matrix a = [ {ll} {2}&{3} {1}&{2} ] satisfies the equation A3 –…
- If a = [ {ll} {3}&{-2} {4}&{-2} ] find k so that A2 = kA – 2I.
- a = [ {lll} {1}&{2}&{5} {0}&{1}&{3} ] , b = [ {lll} {2}&{3}&{0} {1}&{0}&{4}…
- If a = [ {cc} {-1}&{2} {3}&{1} ] find f(A), where f(x) = x2 – 2x + 3.…
- If a = [ {cc} {1}&{2} {4}&{-3} ] and f(x) = 2x3 + 4x + 5, find f(A).…
- Find the values of x and y, when [ {cc} {2}&{-3} {1}&{1} ] [ {x} {y} ] = [ {1}…
- Solve for x and y, when [ {cc} {3}&{-4} {1}&{2} ] [ {x} {y} ] = [ {c} {3} {11}…
- If a = [ {ll} {3}&{1} {7}&{5} ] find x and y such that A2 + xI = yA.…
- If a = [ {ll} {3}&{2} {1}&{1} ] find the value of a and b such that A2 + aA +…
- Find the matrix A such that [ {cc} {5}&{-7} {-2}&{3} ] c. a = [ {rr}…
- Find the matrix A such that A. [ {cc} {2}&{3} {4}&{5} ] = [ {cc} {0}&{-4}…
- If a = [ {ll} {1}&{-1} {2}&{-1} ] , b = [ {ll} {a}&{-1} {b}&{-1} ] and (A +…
- If f (x) = [ {ccc} {cosx}&{-sinx}&{0} {sinx}&{cosx}&{0} {0}&{0}&{1} ] show…
- If a = [ {cc} {cosalpha }&{sinalpha} { - sinalpha , } &{cosalpha} ] show that…
- If [ {cc} {1}&{x} ] [ {ccc} {1}&{2}&{3} {4}&{5}&{6} {3}&{2}&{5} ] [ {c} {1}…
- If [ {ccc} {x}&{4} ] [ {ccc} {2}&{1}&{2} {1}&{0}&{2} {0}&{2}&{-4} ] [ {c} {x}…
- Find the values of a and b for which [ {cc} {a}&{b} {-a}&{2b} ] [ {c} {2} {-1}…
- If a = [ {cc} {3}&{4} {-4}&{-3} ] find f(A), where f(x) = x2 – 5x + 7.…
- If a = [ {ll} {1}&{1} {0}&{1} ] prove that a^{n} = [ {ll} {1}&{n} {0}&{1} ]…
- Given an example of two matrices A and B such thatA ≠ O, B ≠ O, AB = O and BA ≠…
- Give an example of three matrices A, B, C such thatAB = AC but B ≠ C.…
- If a = [ {cc} {1}&{0} {-1}&{7} ] and b = [ {cc} {0}&{4} {-1}&{7} ] find…
- If [ {cc} {2}&{3} {5}&{7} ] [ {cc} {1}&{-3} {-2}&{4} ] = [ {cc} {-4}&{6}…
- If a = [ {rrr} {2}&{-3}&{5} {0}&{7}&{-4} ] verify that (A’)’ = A.…
- If a = [ {rr} {3}&{5} {-2}&{0} {4}&{-6} ] verify that (2A)’ = 2A’.…
- If a = [ {rrr} {3}&{2}&{-1} {-5}&{0}&{-6} ] and b = [ {ccc} {-4}&{-5}&{-2}…
- If p = [ {rr} {3}&{4} {2}&{-1} {0}&{5} ] and p = [ {rr} {7}&{-5} {-4}&{0}…
- If a = [ {ll} {4}&{1} {5}&{8} ] show that (A + A’) is symmetric.…
- If a = [ {ll} {3}&{-4} {1}&{-1} ] show that (A + A’) is skew-symmetric.…
- Show that the matrix a = [ {ccc} {0}&{a}&{b} {-a}&{0}&{c} {-b}&{-c}&{0} ] is…
- Express the matrix a = [ {cc} {2}&{3} {-1}&{4} ] as the sum of a symmetric…
- Express the matrix a = [ {ll} {3}&{-4} {1}&{-1} ] as the sum of a symmetric…
- Express the matrix a = [ {ccc} {-1}&{5}&{1} {2}&{3}&{4} {7}&{0}&{9} ] as the…
- Express the matrix A as the sum of a symmetric and a skew-symmetric matrix,…
- Express the matrix a = [ {lll} {3}&{2}&{5} {4}&{1}&{3} {0}&{6}&{7} ] as sum…
- a = [ {ll} {1}&{3} {2}&{4} ] and b = [ {ll} {1}&{4} {2}&{5} ] For each of…
- a = [ {ll} {3}&{-1} {2}&{-2} ] and b = [ {ll} {1}&{-3} {2}&{-1} ] For each…
- a = [ {c} {-1} {2} {3} ] and B = [-2 -1 -4] For each of the following pairs…
- a = [ {rrr} {-1}&{2}&{-3} {4}&{-5}&{6} ] and b = [ {cc} {3}&{-4} {2}&{1}…
- If a = [ {cc} {cosalpha }&{sinalpha} {-sinalpha}&{cosalpha} ] show that A’A =…
- If matrix A = [1 2 3], write AA’.
- [ {ll} {1}&{2} {3}&{7} ] Using elementary row transformations, find the inverse…
- [ {cc} {1}&{2} {2}&{-1} ] Using elementary row transformations, find the inverse…
- [ {rr} {2}&{5} {-3}&{1} ] Using elementary row transformations, find the inverse…
- [ {cc} {2}&{-3} {4}&{5} ] Using elementary row transformations, find the inverse…
- [ {ll} {4}&{0} {2}&{5} ] Using elementary row transformations, find the inverse…
- [ {ll} {6}&{7} {8}&{9} ] Using elementary row transformations, find the inverse…
- [ {lll} {0}&{1}&{2} {1}&{2}&{3} {3}&{1}&{1} ] Using elementary row…
- [ {ccc} {2}&{-3}&{3} {2}&{2}&{3} {3}&{-2}&{2} ] Using elementary row…
- [ {lll} {3}&{0}&{2} {1}&{5}&{9} {6}&{4}&{7} ] Using elementary row…
- [ {ccc} {1}&{2}&{-3} {2}&{3}&{2} {3}&{-3}&{-4} ] Using elementary row…
- [ {ccc} {3}&{-1}&{-2} {2}&{0}&{-1} {3}&{-5}&{0} ] Using elementary row…
- [ {ccc} {1}&{3}&{-2} {-3}&{0}&{-1} {2}&{1}&{0} ] Using elementary row…
- [ {ccc} {1}&{2}&{3} {2}&{5}&{7} {-2}&{-4}&{-5} ] Using elementary row…
- [ {ccc} {3}&{0}&{-1} {2}&{3}&{0} {0}&{4}&{1} ] Using elementary row…
- [ {ccc} {-1}&{1}&{2} {1}&{2}&{3} {3}&{1}&{1} ] Using elementary row…
- Construct a 3 × 2 matrix whose elements are given by a_{ij} = {1}/{2}…
- Construct a 2 × 3 matrix whose elements are given by a_{ij} = {1}/{2} |-3i+j|…
- If [ {cc} {x+2y}&{-y} {3x}&{4} ] = [ {cc} {-4}&{3} {6}&{4} ] find the values…
- Find the values of x and y, if 2 [ {ll} {1}&{3} {0}&{x} ] + [ {ll} {y}&{0}…
- If x c. [ {2} {3} ]+y [ {c} {-1} {1} ] = [ {c} {10} {5} ] find the values…
- If [ {cc} {x}&{3x-y} {2x+z}&{3y-w} ] = [ {ll} {3}&{2} {4}&{7} ] find the…
- If [ {cc} {x}&{6} {-1}&{2w} ] + [ {cc} {4}&{x+y} {z+w}&{3} ] = 3 [ {ll} {x}&{y}…
- If A = diag (3 -2, 5) and B = diag (1 3 -4), find (A + B).
- Show that
- If a = [ {rr} {1}&{-5} {-3}&{2} {4}&{-2} ] and b = [ {cc} {3}&{1} {2}&{-1}…
- If a = [ {cc} {cosalpha }&{-sinalpha} {sinalpha}&{cosalpha} ] then find the…
- Find the value of x and y for which [ {cc} {2}&{-3} {1}&{1} ] [ {x} {y} ] = […
- Find the value of x and y for which [ {cc} {x}&{y} {3y}&{x} ] [ {1} {2} ] = […
- If a = [ {ll} {4}&{5} {1}&{8} ] show that (A + A’) is symmetric…
- If a = [ {ll} {2}&{3} {4}&{5} ] and show that (A – A’) is skew-symmetric…
- If a = [ {cc} {2}&{-3} {4}&{5} ] and b = [ {cc} {-1}&{2} {0}&{3} ] find a…
- If a = [ {ll} {4}&{2} {1}&{3} ] and b = [ {cc} {-2}&{1} {3}&{2} ] find a…
- If a = [ {cc} {cosalpha }&{sinalpha} {-sinalpha}&{cosalpha} ] show that A’ A…
- If A and B are symmetric matrices of the same order, show that (AB – BA) is a…
- If a = [ {ll} {2}&{3} {1}&{2} ] and f(x) = x2 – 4x + 1, find f(A).…
- If the matrix A is both symmetric and skew-symmetric, show that A is a zero…
Exercise 5a
Question 1.If 
then write
i. the number of rows in A,
ii. the number of columns in A,
iii. the order of the matrix A,
iv. the number of all entries in A,
v. the elements a23, a31, a14, a33, a22 of A.
Answer:
(i) Number of rows = 3
(ii) Number of columns = 4
(iii) Order of matrix = Number of rows x Number of columns = (3 x 4)
(iv) Number of entries = (Number of rows) x (Number of columns)
= 3 x 4
= 12
(V) 
Question 2.
Write the order of each of the following matrices:
i. 
ii. 
iii. 
iv. D = [8 -3]
v. 
vi, F = [6]
Answer:
i.
Order of matrix = Number of rows x Number of columns
= (2 x 4)
ii.
Order of matrix = Number of rows x Number of columns
= (4 x 2)
iii.
Order of matrix = Number of rows x Number of columns
= (1 x 4)
iv. D = [8 -3]
Order of matrix = Number of rows x Number of columns
= (1 x 2)
v.
Order of matrix = Number of rows x Number of columns
= (3 x 1)
vi, F = [6]
Order of matrix = Number of rows x Number of columns
= (1 x 1)
Question 3.
If a matrix has 18 elements, what are the possible orders it can have?
Answer:
Number of entries = (Number of rows) x (Number of columns) = 18
If order is (a x b) then, Number of entries = a x b
So now a x b = 18 (in this case)
Possible cases are (1 x 18), (2 x 9), (3 x 6), (6 x 3), (9 x 2), (18 x 1)
Conclusion: If a matrix has 18 elements, then possible orders are (1 x 18), (2 x 9), (3 x 6), (6 x 3), (9 x 2), (18 x 1)
Question 4.
Find all possible orders of matrices having 7 elements.
Answer:
Number of entries = (Number of rows) x (Number of columns) = 7
If order is (a x b) then, Number of entries = a x b
So now a x b = 7 (in this case)
Possible cases are (1 x 7), (7 x 1)
Conclusion: If a matrix has 18 elements, then possible orders are (1 x 7), (7 x 1)
Question 5.
Construct a 3 × 2 matrix whose elements are given by aij = (2i – j).
Answer:
Given: aij = (2i – j)
Now, a11 = (2 × 1 – 1) = 2 – 1 = 1
a12 = 2 × 1 – 2 = 2 – 2 = 0
a21 = 2 × 2 – 1 = 4 – 1 = 3
a22 = 2 × 2 – 2 = 4 – 2 = 2
a31 = 2 × 3 – 1 = 6 – 1 = 5
a32 = 2 × 3 – 2 = 6 – 2 = 4
Therefore,
Question 6.
Construct a 4 × 3 matrix whose elements are given by 
Answer:
It is (4 x 3) matrix. So it has 4 rows and 3 columns
Given
So, 
, 
, 
,
, 
, 
,
, 
, 
,
, 
, 
So, the matrix 
Conclusion: Therefore, Matrix is 
Question 7.
Construct a 2 × 2 matrix whose elements are 
Answer:
It is a (2 x 2) matrix. So, it has 2 rows and 2 columns.
Given 
So, 
, 
,
, 
So, the matrix 
Conclusion: Therefore, Matrix is
Question 8.
Construct a 2 × 3 matrix whose elements are 
Answer:
It is a (2 x 3) matrix. So, it has 2 rows and 3 columns.
Given 
So, 
, 
, 
,
, 
, 
So, the matrix 
Conclusion: Therefore, Matrix is 
Question 9.
Construct a 3 × 4 matrix whose elements are given by 
Answer:
It is a (3 x 4) matrix. So, it has 3 rows and 4 columns.
Given 
So, , , 
, 
,
, , 
,
,
, 
, 
,
So, the matrix 
Conclusion: Therefore, Matrix is 
Exercise 5b
Question 1.If 
and 
verify that (A + B) = (B + A).
Answer:
A + B 
B + A 
= B + A
Therefore, A + B = B + A
This is true for any matrix
Conclusion: A + B = B + A
Question 2.
If 
and 
verify that (A + B) + C = A + (B+C).
Answer:
(A+B)+C 
A+(B+C) 
Therefore, (A+B)+C = A+(B+C)
It is true for any matrix
Conclusion: (A+B)+C = A+(B+C)
Question 3.
If 
and 
find (2A – B).
Answer:
2A 
(2A-B) 
Conclusion: (2A-B)
Question 4.
Let 
and 
Find:
i. A + 2B
ii. B – 4c
iii. A – 2B + 3C
Answer:
A + 2B 
Conclusion: (A+2B) = 
ii. B – 4c
B-4C 
Conclusion: B-4C
iii. A – 2B + 3C
A-2B+3C 
Conclusion: A_2B+3C
Question 5.
Let 
and 
Compute 5A – 3B + 4C.
Answer:
5A-3B+4C 
Conclusion: 5A-3B+4C
Question 6.
If 
find A.
Answer:
5A 
A 
A 
Conclusion: A
Question 7.
Find matrices A and B, if 
and 
Answer:
Add (A+B) and (A-B)
We get (A+B)+(A-B) 
2A 
A 
Now Subtract (A-B) from (A+B)
(A+B)-(A-B) 
(2B) 
B 
Conclusion: A , B
Question 8.
Find matrices A and B, if 
and 
Answer:
Add 2(2A-B) and (2B+A)
2(2A-B)+(2B+A) 
5A 
5A 
A 
B 
B 
Conclusion: A , B
(GIVEN ANSWER IS WRONG for question 8)
Question 9.
Find matrix X, if 
Answer:
Given 
Conclusion : x
Question 10.
If 
and 
find a matrix C such that A + B – C = O.
Answer:
Given A + B – C = 0
Conclusion:
Question 11.
Find the matrix X such that 2A – B + X = O,
where 
and 
Answer:
Given 2A – B + X = 0
Conclusion: 
Question 12.
If 
and 
find a matrix C such that (A + B + C) is a zero matrix.
Answer:
Given A+B+C is zero matrix i.e A+B+C = 0
Conclusion: 
Question 13.
If A = diag [2, -5, 9], B = diag [-3, 7, 14] and C = diag [4, -6, 3], find:
(i) A + 2B
(ii)B + C – A
Answer:
If Z = diag[a,b,c], then we can write it as
So, A+2B 
=diag[4,9,37]
Conclusion: A + 2B = diag[4,9,37]
(Given answer is wrong)
ii. B + C – A
If Z = diag[a,b,c], then we can write it as
B+C-A 
= diag[-1,6,8]
Conclusion: B+C-A = diag[-1,6,8]
iii. 2A + B – 5C
If Z = diag[a,b,c], then we can write it as
2A+B-5C 
= diag[-19,27,17]
Conclusion: 2A + B – 5C = diag[-19,27,17]
(Given answer is wrong)
Question 14.
Find the value of x and y, when
i. 
Answer:
If 
,
Then a=e, b=f, c=g, d=h
Given 
So, x + y = 8 and x - y = 4
Adding these two gives 2x = 12
y =2
Conclusion : x = 6 and y =2
ii. 
Given, 
So, 2x+5 = x-3 and 3y-7 = -5
Conclusion : x = -8 and y = 
iii. 
2x+3 = 7 
2y-4 = 14 
Conclusion : x = 2 and y = 9
(Given answer is wrong)
Question 15.
Find the value of (x + y) from the following equation :
Answer:
Given
So, 2+y = 5 and 2x+2 = 8
i.e y = 3 and x = 3
Therefore, x+y=6
Conclusion: Therefore x+y = 6
Question 16.
If 
then write the value of (x + y).
Answer:
If ,
Then a=e, b=f, c=g, d=h
Given, 
,
So, x-y = 1, x+y =5, 2y = 4 and 2y+z = 9
Therefore, x+y = 5
Conclusion: x+y = 5
(Given answer is wrong)
Exercise 5c
Question 1.Compute AB and BA, which ever exists when
and 
Answer:
Given : and
Matrix A is of order 3 
2, and Matrix B is of order 2 2
To find : matrix AB and BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = c = 2,d = 2 ,thus matrix AB is of order 3 2
Matrix AB = 
Matrix AB = 
Matrix AB = 
Matrix AB =
For matrix BA, a = 3,b = c = 2,d = 2 ,thus matrix BA exists, if and only if d=a
But 3 
2
Thus matrix BA does not exist
Question 2.
Compute AB and BA, which ever exists when
and 
Answer:
Given : and
Matrix A is of order 3 2, and Matrice B is of order 3 3
To find : matrix AB and BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrice of order c d ,then matrice AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrice of order c d ,then matrice BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = 2,c = 3,d = 3 ,thus matrix AB does not exist, as d a
But 2 3
Thus matrix AB does not exist
For matrix BA, a = 3,b = 2,c = 3,d = 3 ,thus matrix BA is of order 3 2
as d = a = 3
Matrix BA = 
Matrix BA = 
Matrix 
Matrix BA = 
Question 3.
Compute AB and BA, which ever exists when
and 
Answer:
Given : and
Matrix A is of order 2 3 and Matrix B is of order 3 2
To find : matrices AB and BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 2,b = 3,c = 3,d = 2 ,matrix AB exists and is of order 2 2,as
b = c = 3
Matrix AB = 
Matrix AB = 
= 
Matrix AB =
Matrix AB =
For matrix BA, a = 2,b = 3,c = 3,d = 2 ,matrix BA exists and is of order 3 3,as
d = a = 2
Matrix BA = 
Matrix BA = 
Matrix BA = 
Matrix BA =
Question 4.
Compute AB and BA, which ever exists when
A = [1 2 3 4] and 
Answer:
Given : A = [1 2 3 4] and
Matrix A is of order 1 4 and Matrix B is of order 4 1
To find : matrices AB and BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 1,b = 4,c = 4,d = 1 ,matrix AB exists and is of order 1 1,as
b = c = 4
Matrix AB = 
Matrix AB = 
= 
Matrix AB =
Matrix AB =
For matrix BA, a = 1,b = 4,c = 4,d = 1 ,matrix BA exists and is of order 4 4,as
d = a = 1
Matrix BA = 
Matrix BA = 
Matrix BA =
Question 5.
Compute AB and BA, which ever exists when
and 
Answer:
Given : and
Matrix A is of order 3 2 and Matrix B is of order 2 3
To find : matrices AB and BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = 2,c = 2,d = 3 ,matrix AB exists and is of order 3 3,as
b = c = 2
Matrix AB = 
Matrix AB = 
= 
Matrix AB =
Matrix AB =
For matrix BA, a = 3,b = 2,c = 2,d = 3 ,matrix BA exists and is of order 2 2,as
d = a = 3
Matrix BA = 
Matrix BA = 
= 
Matrix BA =
Matrix BA =
Question 6.
Show that AB ≠ BA in each of the following cases :
and 
Answer:
Given : and
Matrix A is of order 2 2 and Matrix B is of order 2 2
To show : matrix AB BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 2,b = c = 2,d = 2 ,thus matrix AB is of order 2 2
Matrix AB = 
Matrix AB = 
= 
Matrix AB =
For matrix BA, a = 2,b = c = 2,d = 2 ,thus matrix BA is of order 2 2
Matrix BA= 
Matrix BA = 
= 
Matrix BA =
Matrix BA = and Matrix AB =
Matrix AB BA
Question 7.
Show that AB ≠ BA in each of the following cases :
and 
Answer:
Given : and
Matrix A is of order 3 3, and Matrix B is of order 3 3
To show : matrix AB BA
The formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix AB =
Matrix AB = 
= 
Matrix AB =
For matrix BA, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix BA=
=
Matrix BA = 
= 
Matrix BA =
Matrix BA = and Matrix AB =
Matrix AB BA
Question 8.
Show that AB = BA in each of the following cases:
and 
Answer:
Given : and
Matrix A is of order 2 2 and Matrix B is of order 2 2
To show : matrix AB = BA
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 2,b = c = 2,d = 2 ,thus matrix AB is of order 2 2
Matrix AB =
Matrix AB = 
Matrix AB =
For matrix BA, a = 2,b = c = 2,d = 2 ,thus matrix BA is of order 2 2
Matrix BA=
Matrix BA =
Matrix BA = Matrix AB =
Thus Matrix AB = BA
Question 9.
Show that AB = BA in each of the following cases:
and 
Answer:
Given : and
Matrix A is of order 3 3 and Matrix B is of order 3 3
To show : matrix AB BA
Formula used :
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix AB =
=
Matrix AB = 
= 
Matrix AB =
For matrix BA, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix BA=
=
Matrix BA = 
= 
Matrix AB BA
Question 10.
Show that AB = BA in each of the following cases:
and 
Answer:
Given : and
Matrix A is of order 3 3 and Matrix B is of order 3 3
To show : matrix AB = BA
Formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix AB =
=
Matrix AB = 
= 
Matrix AB =
For matrix BA, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix BA = 
Matrix BA = 
Matrix BA = 
Matrix BA =
Matrix AB = Matrix BA =
Question 11.
If 
and 
shown that AB = A and BA = B.
Answer:
Given : and 
Matrix A is of order 3 3 and Matrix B is of order 3 3
To show : matrix AB = A, BA = B
Formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix AB, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix AB = 
=
Matrix AB = 
= 
Matrix AB = = Matrix A
Matrix AB = Matrix A
For matrix BA, a = 3,b = c = 3,d = 3 ,thus matrix AB is of order 3 3
Matrix BA = 
Matrix BA = 
Matrix BA = 
= 
Matrix BA = = Matrix B
Matrix BA = = Matrix B
MATRIX AB = A and MATRIX BA = B
Question 12.
If 
and 
, show that AB is a zero matrix.
Answer:
Given : 
and
Matrix A is of order 3 3 , matrix B is of order 3 3 and matrix C is of order 3 3
To show : AB is a zero matrix
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
= 0 matrix
Hence, Proved
Question 13.
For the following matrices, verify that A(BC) = (AB)C :
and C = [1 -2]
Answer:
Given : and C = [1 -2]
Matrix A is of order 2 3 , matrix B is of order 3 1 and matrix C is of order 1 2
To show : matrix A(BC) = (AB)C
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix BC, a = 3,b = c = 1,d = 2 ,thus matrix BC is of order 3 2
Matrix BC = 
= 
= 
Matrix BC =
For matrix A(BC),a = 2 ,b = c = 3 ,d = 2 ,thus matrix A(BC) is of order 2 x 2
Matrix A(BC) = 
= 
Matrix A(BC) = 
= 
Matrix A(BC) =
Matrix A(BC) =
For matrix AB, a = 2,b = c = 3,d = 1 ,thus matrix BC is of order 2 1
Matrix AB =
= 
Matrix AB = 
= 
Matrix AB =
For matrix (AB)C, a = 2,b = c = 1,d = 2 ,thus matrix (AB)C is of order 2 2
Matrix (AB)C = 
= 
=
Matrix (AB)C =
Matrix A(BC) = (AB)C =
Question 14.
Verify that A(B + C) = (AB + AC), when
and 
Answer:
Given : and
Matrix A is of order 2 2 , matrix B is of order 2 2 and matrix C is of order 2 2
To verify : A(B + C) = (AB + AC)
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
B + C = 
+ 
= 
= 
B + C =
Matrix A(B + C) is of order 2 x 2
A(B + C) = 
= 
A(B + C) = 
= 
A(B + C) = 
For matrix AB, a = b = c = d = 2 ,matrix AB is of order 2 x 2
Matrix AB = 
= 
Matrix AB = 
= 
Matrix AB =
For matrix AC, a = b = c = d = 2 ,matrix AC is of order 2 x 2
Matrix AC = 
= 
Matrix AC = 
= 
Matrix AC =
Matrix AB + AC = + = 
=
Matrix AB + AC = A(B + C) =
A(B + C) = (AB + AC)
Question 15.
Verify that A(B + C) = (AB + AC), when
and 
Answer:
Given : and
Matrix A is of order 3 2 , matrix B is of order 2 2 and matrix C is of order 2 2
To verify : A(B + C) = (AB + AC)
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
B + C = 
+ 
= 
= 
B + C =
For Matrix A(B + C), a = 3,b = c = d = 2,thus matrix A(B + C) is of order 3 x 2
A(B + C) = 
= 
A(B + C) = 
= 
A(B + C) =
For matrix AB, a = 3, b = c = d = 2 ,matrix AB is of order 3 x 2
Matrix AB = 
= 
Matrix AB = 
= 
Matrix AB =
For matrix AC, a = 3, b = c = d = 2 ,matrix AC is of order 3 x 2
Matrix AC = 
= 
Matrix AC = 
= 
Matrix AC =
Matrix AB + AC = + = 
=
Matrix AB + AC = A(B + C) =
A(B + C) = (AB + AC)
Question 16.
If 
and 
verify that A(B – C) = (AB – AC).
Answer:
Given : and
Matrix A is of order 3 3; matrix B is of order 3 3 and matrix C is of order 3 3
To verify : A(B – C) = (AB – AC).
The formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
B - C = 
= 
B - C = 
For Matrix A(B - C), a = 3,b = c = d = 3,thus matrix A(B + C) is of order 3 x 3
A(B - C) = 
A(B - C) = 
A(B - C) = 
= 
A(B - C) = 
For matrix AB, a = 3, b = c = d = 3 ,matrix AB is of order 3 x 3
Matrix AB = 
Matrix AB = 
Matrix AB = 
= 
Matrix AB =
For matrix AC, a = 3, b = c = d = 3 ,matrix AC is of order 3 x 3
Matrix AC = 
Matrix AC = 
Matrix AC = 
= 
Matrix AC =
Matrix AB - AC = 
= 
Matrix AB - AC =
A(B – C) = (AB – AC) =
Question 17.
If 
show that A2 = O.
Answer:
Given :
Matrix A is of order 2 2
To show : A2 = O
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
= 
A2 = 
= 
A2 = 
A2 = O
Question 18.
If 
show that A2 = A.
Answer:
Given : 
Matrix A is of order 3 3
To show : A2 = A
Formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
A2 = 
A2 = 
=
A2 = A = 
Question 19.
If 
show that A2 = I.
Answer:
Given :
Matrix A is of order 3 3
To show : A2 = I
Formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
A2 = 
A2 = 
=
A2 = I =
Question 20.
If 
and 
find (3A2 – 2B + I).
Answer:
Given : and
Matrix A is of order 2 2, Matrix B is of order 2 2
To find : 3A2 – 2B + I
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
= 
= 
A2 = 
3A2 = 3 × = 
3A2 =
2B = 2 × 
= 
2B = 
I = 
3A2 – 2B + I = 
+ = 
3A2 – 2B + I = 
3A2 – 2B + I =
Question 21.
If 
then find (-A2 + 6A).
Answer:
Given :
Matrix A is of order 2 2.
To find : -A2 + 6A
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
= 
= 
A2 = 
-A2 = 
= 
6A = 6 × 
= 
6A =
-A2 + 6A = + = 
= 
-A2 + 6A =
Question 22.
If 
show that (A2 – 5A + 7I) = O.
Answer:
Given :
Matrix A is of order 2 2.
To show : A2 - 5A +7I = 0
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 = 
= 
= 
A2 = 
5A = 5 × 
= 
5A =
I =
7I = 
= 
7I =
A2 - 5A + 7I = 
+ = 
=
A2 - 5A + 7I = 0
Question 23.
Show that the matrix 
satisfies the equation A3 – 4A2 + A = O.
Answer:
Given :
Matrix A is of order 2 
2.
To show : A3 - 4A2 + A = 0
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A2 and A3 are matrices of order 2 x 2.
A2 = 
= 
= 
A2 = 
A3 = 
= 
= 
A3 = 
4A2 = 4 ×
= 
4A2 =
A3 - 4A2 + A = 
+ 
= 
=
A3 - 4A2 + A = 0
Question 24.
If 
find k so that A2 = kA – 2I.
Answer:
Given : A2 = kA – 2I.
Matrix A is of order 2 2.
To find : k
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A2 is a matrix of order 2 x 2.
A2 = 
= 
= 
A2 = 
kA = k × 
= 
kA – 2I = 
= 
It is the given that A2 = kA – 2I
=
Equating like terms,
3k – 2 = 1
3k = 1 + 2 = 3
3k = 3
k = 
= 1
k = 1
Question 25.
For the following matrices, verify that A(BC) = (AB)C :
and 
Answer:
Given : and
Matrix A is of order 2 3 , matrix B is of order 3 3 and matrix C is of order 3 1
To show : matrix A(BC) = (AB)C
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
If A is a matrix of order a b and B is a matrix of order c d ,then matrix BA exists and is of order c b ,if and only if d = a
For matrix BC, a = 3,b = c = 3,d = 1 ,thus matrix BC is of order 3 1
Matrix BC = 
= 
= 
Matrix BC = 
For matrix A(BC),a = 2 ,b = c = 3 ,d = 1 ,thus matrix A(BC) is of order 2 x 1
Matrix A(BC) = 
= 
= 
Matrix A(BC) = 
Matrix A(BC) =
For matrix AB, a = 2,b = c = 3,d = 3 ,thus matrix BC is of order 2 3
Matrix AB =
Matrix AB =
Matrix AB = 
= 
Matrix AB =
For matrix (AB)C, a = 2,b = c = 3,d = 1 ,thus matrix (AB)C is of order 2 1
Matrix (AB)C = 
= 
Matrix (AB)C = 
=
Matrix (AB)C =
Matrix A(BC) = (AB)C =
Question 26.
If 
find f(A), where f(x) = x2 – 2x + 3.
Answer:
Given : and f(x) = x2 – 2x + 3.
Matrix A is of order 2 2.
To find : f(A)
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A2 is a matrix of order 2 x 2.
f(x) = x2 – 2x + 3
f(A) = A2 – 2A + 3I
A2 = 
= 
A2 = 
=
A2 =
2A = 2 × 
= 
2A =
3I = 3 × = 
3I =
f(A) = A2 – 2A + 3I = 
+ = 
f(A) = A2 – 2A + 3I = 
f(A) = A2 – 2A + 3I =
Question 27.
If 
and f(x) = 2x3 + 4x + 5, find f(A).
Answer:
Given : and f(x) = 2x3 + 4x + 5
Matrix A is of order 2 2.
To find : f(A)
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A3 is a matrix of order 2 x 2.
f(x) = 2x3 + 4x + 5
f(A) = 2A3 + 4A + 5I
A2 = 
= 
= 
A2 = 
A3 = × 
= 
A3 = 
= 
A3 = 
2A3 = 2 ×
= 
2A3 =
4A = 4 × = 
4A =
5I = 5 × = 
5I = 
2A3 + 4A + 5I = + + = 
f(A) = 2A3 + 4A + 5I = 
f(A) = 2A3 + 4A + 5I = 
Question 28.
Find the values of x and y, when
Answer:
Given :
To find : x and y
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
The resulting matrix obtained on multiplying 
and 
is of order 2 × 1
× = 
= 
=
Equating similar terms,
2x – 3y = 1 equation 1
x + y = 3 equation 2
equation 1 + 3(equation 2) and solving the above equations,
x = 
= 2
x = 2 , substituting x = 2 in equation 2,
2 + y = 3
y = 3 – 2 = 1
x = 2 and y = 1
Question 29.
Solve for x and y, when
Answer:
Given :
To find : x and y
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
The resulting matrix obtained on multiplying 
and is of order 2 × 1
× = 
= 
=
Equating similar terms,
3x – 4y = 3 equation 1
x + 2y = 11 equation 2
equation 1 + 2(equation 2) and solving the above equations,
5x = 25
x = 
= 5
x = 5 , substituting x = 2 in equation 2,
5 + 2y = 11
2y =11 – 5 = 6
2y = 6
y = 
= 3
x = 5 and y = 3
Question 30.
If 
find x and y such that A2 + xI = yA.
Answer:
Given : A2 + xI = yA.
A is a matrix of order 2 x 2
To find : x and y
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 is a matrix of order 2 x 2
A2 = 
= 
= 
A2 = = 
A2 =
xI = 
= 
xI =
A2 + xI = + = 
= 
A2 + xI =
yA = y 
= 
yA =
It is given that A2 + xI = yA,
=
Equating similar terms in the given matrices,
16 + x = 3y and 8 = y,
hence y = 8
Substituting y = 8 in equation 16 + x = 3y
16 + x = 3 × 8 = 24
16 + x = 24
x = 24 – 16 = 8
x = 8
x = 8, y = 8
Question 31.
If 
find the value of a and b such that A2 + aA + bI = O.
Answer:
Given : A2 + aA + bI = O
A is a matrix of order 2 x 2
To find : a and b
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A2 is a matrix of order 2 x 2
A2 = 
= 
= 
A2 = 
aA = a
= 
bI = b 
= 
bI =
A2 + aA + bI = + + = 
A2 + aA + bI = 
It is given that A2 + aA + bI = 0
=
Equating similar terms in the matrices,we get
4 + a = 0 and 3 + a + b = 0
a = 0 – 4 = -4
a = -4
substituting a = -4 in 3 + a + b = 0
3 – 4 + b = 0
-1 + b = 0
b = 0 + 1 = 1
b = 1
a = -4 and b = 1
Question 32.
Find the matrix A such that 
Answer:
Given :
To find : matrix A
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
IF XA = B, then A = X-1B
. A = 
A = 
To find 
Determinant of given matrix = 
= 5(3) – (-7)(-2) = 15 – 14 = 1
Adjoint of matrix = 
= 
× =
=
A = = 
A = 
= 
= 
A =
A =
Question 33.
Find the matrix A such that A. 
Answer:
Given : A.
To find : matrix A
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
IF AX = B, then A = BX-1
A. 
= 
A = 
To find 
Determinant of given matrix = 
= 5(2) – (4)(3) = 10 – 12 = -2
Adjoint of matrix = 
= 
× =
=
A = = 
A = 
= 
A = 
= 
= 
A =
A =
Question 34.
If 
and (A + B)2 = (A2 + B2) then find the values of a and b.
Answer:
Given :
(A + B)2 = (A2 + B2)
To find : a and b
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A + B = 
+ 
= 
= 
A + B =
(A + B)2 = × = 
(A + B)2 = 
= 
(A + B)2 =
A2 = × = 
= 
A2 = 
B2 = × = 
= 
B2 =
(A2 + B2) = + = 
(A2 + B2) =
It is given that (A + B)2 = (A2 + B2)
=
Equating similar terms in the given matrices we get,
2 – 2a = -a + 1 and -2b = -b + 1
2 – 1 = -a + 2a and -2b + b = 1
1 = a and -b = 1
a = 1 and b = -1
Question 35.
If 
show that F(x) . F(y) = F(x + y).
Answer:
Given :
To show : F(x) . F(y) = F(x + y).
Formula used : 
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
F(x) = 
F(y) = 
F(x + y) = 
F(x) . F(y) = .
= 
F(x) . F(y) = 
We know that,
cosx(cosy) – sinx (siny) = cos(x+y) and -cosx(siny) - sinx(cosy) = -sin(x+y)
F(x) . F(y) =
F(x + y) = F(x) . F(y) =
F(x + y) = F(x) . F(y)
Question 36.
If 
show that 
Answer:
Given :
To show :
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
A = 
A2 = ×
A2 = 
A2 = 
We know that cos2α = 
and sin2α = 
A2 = 
A2 =
Question 37.
If 
find x.
Answer:
Given :
To find : x
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
= 0
= 
= 
= 
= × 
× = 
× = 
= 
= = 0
12x + 20 = 0
12x = -20
x = 
= 
x =
Question 38.
If 
find x.
Answer:
Given : 
To find : x
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
= 0
= 
= 
= 
= 
= 
= 0
2x2 + 6x + 4 = 0
x2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
x + 1 = 0 or x + 2 = 0
x = -1 or x = -2
x = -1 or x = -2
Question 39.
Find the values of a and b for which
Answer:
Given :
To find : a and b
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
If A is a matrix of order a b and B is a matrix of order c d ,then matrix AB exists and is of order a d ,if and only if b = c
= 
= 
= 
=
=
Equating similar terms,
2a – b = 5
-2a – 2b = 4
Adding the above two equations,we get
-3b = 9
b = 
= -3
b = -3
substituting b = -3 in 2a – b = 5,we get
2a + 3 = 5
2a = 5 – 3 = 2
a = 1
a = 1 and b = -3
Question 40.
If 
find f(A), where f(x) = x2 – 5x + 7.
Answer:
Given : and f(x) = x2 – 5x + 7.
Matrix A is of order 2 2.
To find : f(A)
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A2 is a matrix of order 2 x 2.
f(x) = x2 – 5x + 7
f(A) = A2 – 5A + 7I
A2 = 
= 
= 
A2 = 
5A = 5 × 
= 
5A =
7I = 7 × =
7I =
f(A) = A2 – 5A + 7I = 
+ = 
f(A) = A2 – 5A + 7I = 
f(A) = A2 – 5A + 7I =
Question 41.
If 
prove that 
for all n ∈ N.
Answer:
Given :
Matrix A is of order 2 2.
To prove :
Proof :
A = 
Let us assume that the result holds for An – 1
An – 1 = 
We need to prove that the result holds for An by mathematical induction .
An = An – 1 × A = 
= 
An = 
= 
An = 
Question 42.
Given an example of two matrices A and B such that
A ≠ O, B ≠ O, AB = O and BA ≠ O.
Answer:
Given : A ≠ 0,B ≠ 0 ,AB = 0, BA ≠ 0
To Find : matrix A and B
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
Let A = 
and B = 
A ≠ 0,B ≠ 0
AB = 
= 
=
AB = = 0
BA = 
= 
=
BA =
A = and B =
Question 43.
Give an example of three matrices A, B, C such that
AB = AC but B ≠ C.
Answer:
Given : AB = AC and B ≠ C.
To Find : matrix A and B
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
Let A = , B = and C = 
B ≠ C
AB = 
= 
AB = = 0
AC = 
= 
AC = = 0
AB = AC = 0
A = , B = and C =
Question 44.
If 
and 
find (3A2 – 2B + I).
Answer:
Given : and
Matrices A and B are of order 2 2.
To find : (3A2 – 2B + I).
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
A2 is a matrix of order 2 x 2.
A = 
A2 = 
= 
A2 = 
= 
3A2 = 3 × = 
3A2 =
2B = 2 × =
2B =
I =
3A2 – 2B + I = 
+ = 
3A2 – 2B + I = 
Question 45.
If 
find the value of x.
Answer:
Given :
To find : x
Formula used :
Where cij = ai1b1j + ai2b2j + ai3b3j + ……………… + ainbnj
= 
= 
= 
= 
= =
=
Equating similar terms in the two matrices, we get
x = 13
x = 13
Exercise 5d
Question 1.If 
verify that (A’)’ = A.
Answer:
Transpose of a matrix is obtained by interchanging the rows and the columns of matrix A. It is denoted by A’.
e.g. A12 = A21
Hence transpose of matrix A is,
(A')' = A
Hence, Proved.
Question 2.
If 
verify that (2A)’ = 2A’.
Answer:
Given 
To Prove: (2A)’ = 2A’
Proof: Let us consider, B = 2A
Now, 
LHS
Again to find RHS, we will find the transpose of matrix A
RHS = 2A’
LHS = RHS
Hence proved.
Question 3.
If 
and 
verify that (A + B)’ = (A’ + B’).
Answer:
Given 
and 
To Prove: (A + B)’ = A’ + B’
Proof: Let us consider C = A + B
Now LHS = C’
To find RHS, we will find transpose of matrix A and B
And 
RHS = A’ + B’
LHS = RHS
Hence proved.
Question 4.
If 
and 
verify that (P + Q)’ = (P’ + Q’).
Answer:
Given 
and 
To Prove: (P + Q)’ = P’ + Q’
Proof: Let us consider R = P + Q,
LHS = R 
(P + Q)’
To find RHS, we will first find the transpose of matrix P and Q
And 
RHS = P’ + Q’
LHS = RHS
Hence proved.
Question 5.
If 
show that (A + A’) is symmetric.
Answer:
Given 
To Prove: A + A’ is symmetric.(Note:A matrix P is symmetric if P’ = P)
Proof: We will find A’,
Now let us take P = A + A’
Now 
Hence A + A’ is a symmetric matrix.
Question 6.
If 
show that (A + A’) is skew-symmetric.
Answer:
Given 
To prove: A-A’ is a skew-symmetric matrix.(Note: A matrix P is skew-symmetric if P’ = -P)
Proof: First we will find the transpose of matrix A
Let us take P = A-A’
P’ = P
Hence A-A’ is a skew symmetric matrix.
Question 7.
Show that the matrix 
is skew-symmetric.
HINT: Show that A’ = -A.
Answer:
Given 
To Prove: A is a skew symmetric matrix.
Proof: As for a matrix to be skew symmetric A’ = -A
We will find A’.
= -
A’ = -A
So A is A skew symmetric matrix.
Question 8.
Express the matrix 
as the sum of a symmetric matrix and a skew-symmetric matrix.
Answer:
Given 
, As for a symmetric matrix A’ = A hence
A + A’ = 2A
A
(Symmetric Matrix)
Similarly for a skew symmetric matrix since A’ = -A hence
A-A’ = 2A
A
(Skew Symmetric Matrix)
So a matrix can be represented as a sum of a symmetric matrix P and skew symmetric matrix Q.
First, we will find the transpose of matrix A,
Now using the above formulas,
Hence A = P + Q
+ 
[Matrix A as the sum of P and Q]
Question 9.
Express the matrix 
as the sum of a symmetric matrix and a skew-symmetric matrix.
Answer:
Given 
,to express as sthe um of symmetric matrix P and skew symmetric matrix Q.
A = P + Q
Where 
and 
,we will find transpose of matrix A
Now using the above formulas
Hence A = P + Q
[Matrix A as the sum of P and Q]
Question 10.
Express the matrix 
as the sum of a symmetric and a skew-symmetric matrix.
Answer:
Given
, to express as sum of symmetric matrix P and skew symmetric matrix Q.
A = P + Q
Where and
,
First, we find A’
Now using the above mentioned formulas
Now A = P + Q
[Matrix A as sum of P and Q]
Question 11.
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where 
Answer:
Given
, to express as sum of symmetric matrix P and skew symmetric matrix Q
A = P + Q
Where and,
First we will find A’,
A’
Now using above mentioned formulas,
Now A = P + Q
[Matrix A as sum of P and Q]
Question 12.
Express the matrix 
as sum of two matrices such that one is symmetric and the other is skew-symmetric.
Answer:
Given
, to express as sum of symmetric matrix P and skew symmetric matrix Q.
A = P + Q
Where and,
First we will find A’
Now using above mentioned formulas
Now A = P + Q
Question 13.
For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :
and 
Answer:
Let us take C = AB
To find RHS we will find transpose of matrix A and B,
And 
RHS = B’A’
LHS = RHS
Hence proved.
Question 14.
For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :
and 
Answer:
Let us take C = AB
To find RHS we will find transpose of matrix A and B,
And 
RHS = B’A’
LHS = RHS
Hence proved.
Question 15.
For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :
and B = [-2 -1 -4]
Answer:
Let us take C = AB
LHS = C’
To find RHS we will find transpose of matrix A and B,
And 
RHS = B’A’
LHS = RHS
Hence proved.
Question 16.
For each of the following pairs of matrices A and B, verify that (AB)’ = (B’ A’) :
and 
Answer:
Let us take C = AB
LHS = C’
To find RHS we will find transpose of matrix A and B,
And 
RHS = B’A’
LHS = RHS
Hence proved.
Question 17.
If 
show that A’A = I.
Answer:
Given 
, We will find A’
LHS = A’A
[Using 
and commutative law a.b = b.a i.e. 
]
RHS = I 
LHS = RHS
Hence proved.
Question 18.
If matrix A = [1 2 3], write AA’.
Answer:
Given 
[1 2 3]
We will find A’ to calculate AA’,
Now
[1 + 4 + 9]
[14]
Exercise 5e
Question 1.Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I = 
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 2.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A = 
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into the Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 3.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A = 
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 4.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 5.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I = 
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 6.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A = 
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 7.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I = 
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 8.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 9.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 10.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 11.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 12.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 13.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 14.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I =
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Question 15.
Using elementary row transformations, find the inverse of each of the following matrices:
Answer:
Let, A =
Now we are going to write the Augmented Matrix followed by matrix A and the Identity matrix I, i.e.,
, where I = 
Now our job is to convert the matrix A into Identity Matrix. Therefore, the matrix we will get converting the matrix I will be our A-1.
Here, the matrix A is converted into Identity matrix. Therefore, we get the A-1 as,
A-1 = 
[Answer]
The value of A-1 is correct or not can be verified by the formula: AA-1 = I
Exercise 5f
Question 1.Construct a 3 × 2 matrix whose elements are given by
Answer:
Here, i is the subscript for a row, and j is the subscript for column
And the given matrix is 3×2, so 1≤ i ≤ 3 and 1≤j ≤2
Hence for i=1, j=1, 
= 
For i=1, j=2, 
= 
For i=2, j=1 
= 0
For i=2, j=2 
= 2
For i=3, j=1 
=
For i=3, j=2 
=
Hence the required matrix is :-
Question 2.
Construct a 2 × 3 matrix whose elements are given by
Answer:
The elements of the matrix are given by, 
Matrix is 2
hence, 
Here, i is the subscript for a row, and j is the subscript for column
For i=1, j=1, 
For i=1, j=2, 
For i=1, j=3, 
For i=2, j=1, 
For i=2, j=2, 
For i=2, j=3, 
Hence the required matrix is :-
Question 3.
If 
find the values of x and y.
Answer:
On comparing L.H.S. and R. H.S we get,
On comparing each term we get,
….(i)
…(ii)
…..(iii)
From (i), (ii) and (iii), we get,
Question 4.
Find the values of x and y, if
Answer:
Given,
Using the property of matrix multiplication such that h is scalar, 
Using the matrix property of matrix addition, when two matrices are of the same order then, each element gets added to the corresponding element,
Comparing each element we get,
2+y=5, ⇒ y=3
2x+2=8, ⇒x=3
Question 5.
If 
find the values of x and y.
Answer:
Given, 
And we have,
Solving the linear equations, we get,
Question 6.
If 
find the values of x, y, z, ω.
Answer:
Given,
On comparing each element of the two matrices we get,
x=3,
3x-y=2
y=7
2x+z=4,
z=-2,
3y-w=7,
w=14
Question 7.
If 
find the values of x, y, z, ω.
Answer:
Given,
First applying matrix addition then, comparing each element of the matrix with the corresponding element we get,
We now have, 
…..(i)
x=2
…….(ii)
6+x+y=3y, substituting x from (i) we get,
y = 4,
And -1+z+w=3z, substituting w from (ii), we get,
z=1
Question 8.
If A = diag (3 -2, 5) and B = diag (1 3 -4), find (A + B).
Answer:
We are given two diagonal matrices A and B,
On adding the two diagonal matrices of order (3) we get an diagonal matrix of order (3
3)
Each of the elements get added to the corresponding element hence, we get after adding, 
Hence, we get A+B = diag(4 1 1)
Question 9.
Show that
Answer:
We have to show that
Multiplying the scalars with we get,
And we know that 
Hence, proved.
Question 10.
If 
and 
find the matrix C such that A + B + C is a zero matrix
Answer:
Given, A+B+C = zero matrix
We know that zero matrix is a matrix whose all elements are zero, so we have,
WE have A+B+C=0,
So C = -A+B,
Question 11.
If 
then find the least value of α for which A + A’ = I.
Answer:
Given, 
Here, A’ i.e. A transpose is 
We are given that A+A’=I
So, 
After doing addition of matrices, we get,
On comparing the elements we get,
This implies, 
For 
belongs 0 to 
, 
= 
Question 12.
Find the value of x and y for which
Answer:
Given,
Applying matrix multiplication we get,
On comparing the elements we get, 2x-3y = 1,
x+y = 3,
On solving the equations we get, x=2, y=1
Question 13.
Find the value of x and y for which
Answer:
Given,
Applying matrix multiplication we have, 
On comparing the elements with each other we get,
The linear equations, x+2y=3, 3y+2x=5
On solving these equations we get x = 1, y = 1
Question 14.
If 
show that (A + A’) is symmetric
Answer:
Given, 
Then, (A +A’) will be, 
The matrix 
is a symmetrical matrix.
Question 15.
If 
and show that (A – A’) is skew-symmetric
Answer:
Given,
, and
(A - A’) = 
The matrix 
is skew-symmetric.
Question 16.
If 
and 
find a matrix X such that A + 2B + X = O.
Answer:
Given, 
We need to a matrix X such that, A +2B + X = 0
We have, X = -(A + 2B),
Question 17.
If 
and 
find a matrix X such that
3 A – 2B + X = O.
Answer:
Given, 
We have 3A – 2B + X = 0
So X = -(3A – 2B)
Thus,
Question 18.
If 
show that A’ A = I.
Answer:
Given, 
Then , 
Applying matrix multiplication we get,
Hence, 
As we know that 
Question 19.
If A and B are symmetric matrices of the same order, show that (AB – BA) is a skew symmetric matrix.
Answer:
We are given that A and B are symmetric matrices of the same order then, we need to show that (AB – BA) is a skew symmetric matrix.
Let us consider P is a matrix of the same order as A and B
And let P = (AB – BA),
we have A = A’ and B = B’
then, P’ = (AB – BA)’
P’ = ((AB)’ – (BA)’) …….using reversal law we have (CD)’=D’C’
P’ = (B’A’ – A’B’)
P’ = (BA – AB)
P’ = -P
Hence, P is a skew symmetric matrix.
Question 20.
If 
and f(x) = x2 – 4x + 1, find f(A).
Answer:
Given, 
f(x) = x2 – 4x + 1,
f(A) = A2 – 4A + I,
f(A) = 
f(A) = 
f(A) = 
Question 21.
If the matrix A is both symmetric and skew-symmetric, show that A is a zero matrix.
Answer:
Given that matrix A is both symmetric and skew symmetric, then,
We have A = A’ ……(i)
And A = -A’ ……(ii)
From (i) and (ii) we get,
A’ = -A’,
2A’ = 0
A’ = 0
Then, A = 0
Hence proved.