NCERT Solutions Class 12

NCERT Solutions Class 12

Matrices : NCERT Solutions – Class 12 Maths (Ex 3)

Exercise 3.3

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.

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