Ncert Solution of Math Class 12 Chapter 3

NCERT Solutions for Class 12 Maths chapter 3-Matrices Exercise 3.3

NCERT Solutions for Class 12 Maths Chapter-3 Matrices

NCERT solutions for class 12 maths Chapter-3 Matrices is prepared by academic team of Entrancei. We have prepared NCERT Solutions for all exercise of chapter-3. Given below is step by step solutions of all questions given in NCERT textbook for Chapter-3 Matrices.

NCERT Solutions for Class 12 Maths Exercise 3.3

Solve The Following Questions.

Question 1. Find the transpose of each of the following matrices:

(i)

(ii)

(iii)

Solution :
(i) Let A =

  Transpose of A = A' or AT =

(ii)

  Transpose of A = A' or AT =

(iii)

  Transpose of A = A' or AT =

Question 2. If A =  and B =  then verify that:

(i)

(ii)

Solution :
(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.

Question 3. If A' =  and B =  then verify that:

(i)

(ii)

Solution :
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.

Question 4. If A' =  and B =  then find (A + 2B)'.

Solution :
Given: A' =  and B =  then (A')' = A =

A +2B =  =  =  =

  (A + 2B)' =

Question 5. For the matrices A and B, verify that (AB)' = B'A', where:

(i) A =  B =

(ii) A =  B =

Solution :
(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.

Question 6. (i) If A =  then verify that A'A = I.

(ii) If A =  then verify that A'A = I.

Solution :
(i) L.H.S. = A'A =

=

=  =  = I = R.H.S.

(ii) L.H.S. = A'A =  =

=  =  = I = R.H.S.

Question 7. (i) Show that the matrix A =  is a symmetric matrix.

(ii) Show that the matrix A =  is a skew symmetric matrix.

Solution :
(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

Question 8. For a matrix A =  verify that:

(i) (A + A') is a symmetric matrix.

(ii) (A – A') is a skew symmetric matrix.

Solution :
(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.

Question 9. Find  (A + A') and  (A – A') when A =

Solution :
Given: A =   A' =

Now, A + A' =  =  =

 (A + A') =

Now, A – A' =  =  =

 (A – A') =  =

Question 10. Express the following matrices as the sum of a symmetric and skew symmetric matrix:

(i)

(ii)

(iii)

(iv)

Solution :
(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.

Question 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

(D) Identity matrix

Solution :
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.

Question 12. If A = , then A + A' = I, if the value of  is:

(A)

(B)

(C)

(D)

Solution :
Given: A =   Also A + A' = I

Equating corresponding entries, we have

Therefore, option (B) is correct.

Ncert Solution of Math Class 12 Chapter 3

Source: https://www.entrancei.com/ncert-solutions-class-12-maths-chapter-3-exercise-3.3

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