# NCERT Solutions class 12 Maths Exercise 3.3 Ch 3 Matrices

## NCERT Solutions for Class 12 Maths Exercise 3.3 NCERT Solutions for Class 12 Maths Chapter 3 – Matrices – FREE PDF Download

NCERT Class 12 Maths Ch 3 is one of the most important ones in the NCERT syllabus. Duly following NCERT Solutions for Class 12 Maths Chapter 3 ensures you that there will be no hindrance when you opt for more advanced branches of Maths. Class 12 Maths Chapter 3 – Matrices solved by Expert Teachers as per NCERT (CBSE) Book guidelines. All Matrices Exercise Questions with Solutions to help you to revise complete Syllabus and Score More marks.

# NCERT Solutions for Class 12 Maths Chapter 3 – 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.

### 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.

### 4. If A’ =  and B =  then find (A + 2B)’.

Ans. Given: A’ =  and B =  then (A’)’ = A =

A +2B =

(A + 2B)’ =

### 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.

### 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.

### 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’) =

### 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’) =

=

=> A =  + .

(ii) Given: A =      A’ =

Symmetric matrix =  (A + A’) =

=

And Skew symmetric matrix =  (A – A’) =

=

A =  + .

(iii) Given: A =      A’ =

Symmetric matrix =  (A + A’) =

=

And Skew symmetric matrix =  (A – A’) =

=

A =   + .

(iv) Given: A =       A’ =

Symmetric matrix =  (A + A’) =  =  =

And Skew symmetric matrix =  (A – A’) =  =

A = + .

### 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.

### 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.