Home » NCERT Solutions » NCERT Solutions class 12 Maths Exercise 3.3 Ch 3 Matrices

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.