NCERT Solutions class 12 Maths Exercise 3.3

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.

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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 A^T =

(ii)

  Transpose of A = A’ or A^T =

(iii)

  Transpose of A = A’ or A^T =

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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.

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NCERT Solutions class 12 Maths Exercise 3.3

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.

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NCERT Solutions class 12 Maths Exercise 3.3

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

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

A +2B =  =  =  =

  (A + 2B)’ =

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NCERT Solutions class 12 Maths Exercise 3.3

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.

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NCERT Solutions class 12 Maths Exercise 3.3

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.

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NCERT Solutions class 12 Maths Exercise 3.3

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

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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.

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9. Find  (A + A’) and  (A – A’) when A =  

Ans. Given: A =      A’ =

Now, A + A’ =  =  =

   (A + A’) =

Now, A – A’ =  =  =

   (A – A’) =  =

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NCERT Solutions class 12 Maths Exercise 3.3

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 .

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NCERT Solutions class 12 Maths Exercise 3.3

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.

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NCERT Solutions class 12 Maths Exercise 3.3

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

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