Exercise 3.3
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NCERT Solutions class 12 Maths Matrices
1. Find the transpose of each of the following matrices:
(i) 
(ii) 
(iii) 
Ans. (i) Let A =
Transpose of A = A’ or AT = 
(ii)
Transpose of A = A’ or AT = 
(iii)
Transpose of A = A’ or AT = 
2. If A = 
and B = 
then verify that:
(i) 
(ii) 
Ans. (i) A + B = 
= 
= 
L.H.S. = (A + B)’ = 
= 
R.H.S. = A’ + B’ = 
= 
= 
=
L.H.S. = R.H.S. Proved.
(ii) A – B = 
= 
= 
L.H.S. = (A – B)’ = 
= 
R.H.S. = A’ – B’ = 
= 
= 
=
L.H.S. = R.H.S. Proved.
NCERT Solutions class 12 Maths Exercise 3.3
3. If A’ = 
and B = 
then verify that:
(i)
(ii)
Ans. Given: A’ = and B = then (A’)’ = A = 
(i) A + B = 
= 
L.H.S. = (A + B)’ = 
R.H.S. = A’ + B’ = 
= 
= 
= 
L.H.S. = R.H.S. Proved.
(ii) A – B = 
= 
L.H.S. = (A – B)’ = 
R.H.S. = A’ – B’ = 
= 
= 
= 
L.H.S. = R.H.S. Proved.
NCERT Solutions class 12 Maths Exercise 3.3
4. If A’ = 
and B = 
then find (A + 2B)’.
Ans. Given: A’ = and B = then (A’)’ = A = 
A +2B = 
= 
= 
= 
(A + 2B)’ = 
NCERT Solutions class 12 Maths Exercise 3.3
5. For the matrices A and B, verify that (AB)’ = B’A’, where:
(i) A = 
B = 
(ii) A = 
B = 
Ans. (i) AB = 
= 
L.H.S. = (AB)’ = 
= 
R.H.S. = B’A’ = 
= 
=
L.H.S. = R.H.S. Proved.
(ii) AB = 
= 
L.H.S. = (AB)’ = 
= 
R.H.S. = B’A’ = 
= 
=
L.H.S. = R.H.S. Proved.
NCERT Solutions class 12 Maths Exercise 3.3
6. (i) If A = 
then verify that A’A = I.
(ii) If A = 
then verify that A’A = I.
Ans. (i) L.H.S. = A’A = 
= 
= 
= 
= I = R.H.S.
(ii) L.H.S. = A’A = 
= 
= 
= = I = R.H.S.
NCERT Solutions class 12 Maths Exercise 3.3
7. (i) Show that the matrix A = 
is a symmetric matrix.
(ii) Show that the matrix A = 
is a skew symmetric matrix.
Ans. (i) Given: A = ……….(i)
Changing rows of matrix A as the columns of new matrix A’ = = A
A’ = A
Therefore, by definitions of symmetric matrix, A is a symmetric matrix.
(ii) Given: A = ……….(i)
A’ = 
= 
Taking 
common, A’ = 
= – A [From eq. (i)]
Therefore, by definition matrix A is a skew-symmetric matrix
8. For a matrix A = 
verify that:
(i) (A + A’) is a symmetric matrix.
(ii) (A – A’) is a skew symmetric matrix.
Ans. (i) Given: A = 
Let B = A + A’ = 
= 
= 
B’ = 
= B
B = A + A’ is a symmetric matrix.
(ii) Given:
Let B = A – A’ = 
= 
= 
B’ = 
Taking common, 
= – B
B = A – A’ is a skew-symmetric matrix.
9. Find 
(A + A’) and (A – A’) when A = 
Ans. Given: A = 
A’ = 
Now, A + A’ = 
= 
= 
(A + A’) = 
Now, A – A’ = 
= 
= 
(A – A’) = 
= 
NCERT Solutions class 12 Maths Exercise 3.3
10. Express the following matrices as the sum of a symmetric and skew symmetric matrix:
(i) 
(ii) 
(iii) 
(iv) 
Ans. (i) Given: A = A’ = 
Symmetric matrix = (A + A’) = 
= 
= 
And Skew symmetric matrix = (A – A’) = 
= 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(ii) Given: A = A’ =
Symmetric matrix = (A + A’) = 
= 
=
And Skew symmetric matrix = (A – A’) = 
= 
=
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iii) Given: A = A’ = 
Symmetric matrix = (A + A’) = 
= 
= 
And Skew symmetric matrix = (A – A’) = 
= 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
(iv) Given: A = A’ = 
Symmetric matrix = (A + A’) = 
= 
= 
And Skew symmetric matrix = (A – A’) = 
= 
Given matrix A is sum of Symmetric matrix and Skew symmetric matrix .
NCERT Solutions class 12 Maths Exercise 3.3
Choose the correct answer in Exercises 11 and 12.
11. If A and B are symmetric matrices of same order, AB – BA is a:
(A) Skew-symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(S) Identity matrix
Ans. Given: A and B are symmetric matrices 
A = A’ and B = B’
Now, (AB – BA)’ = (AB)’ – (BA)’ (AB – BA)’ = B’A’ – A’B’ [Reversal law]
(AB – BA)’ = BA – AB [From eq. (i)] (AB – BA)’ = – (AB – BA)
(AB – BA) is a skew matrix.
Therefore, option (A) is correct.
NCERT Solutions class 12 Maths Exercise 3.3
12. If A = 
, then A + A’ = I, if the value of 
is:
(A) 
(B) 
(C) 
(D) 
Ans. Given: A = 
Also A + A’ = I
Equating corresponding entries, we have
Therefore, option (B) is correct.