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NCERT Solutions class 12 Maths Exercise 3.3 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

## 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.

#### 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’) =

=  =

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 .

#### 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.

## NCERT Solutions class 12 Maths Exercise 3.3

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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