Class 12 NCERT Solutions - Mathematics Part I - Chapter 3 Matrices - Exercise 3.3

Chapter 3 of the Class 12 NCERT Mathematics textbook, titled "Matrices," delves into the fundamental concepts of matrices, including their types, operations, and applications. Exercise 3.3 focuses on practical problems involving matrix operations, such as addition, subtraction, and multiplication of matrices. This exercise helps students apply their understanding of matrices to solve various mathematical problems.

Mathematics Part I - Chapter 3 Matrices - Exercise 3.3

This section provides detailed solutions for Exercise 3.3 from Chapter 3 of the Class 12 NCERT Mathematics textbook. The exercise includes a variety of problems related to matrix operations, offering step-by-step explanations to ensure students can effectively solve these problems and grasp the underlying concepts of matrix algebra.

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

(i)  

(ii) 

(iii) 

Solution:

(i) Let A =

∴Transpose of A = A' = A^T  = 

(ii) Let A =

∴Transpose of A = A' = AT  =

(iii) Let A =

∴Transpose of A = A' = AT  =

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

(i) (A+B)' = A'+B'

(ii) (A-B)' = A'- B'

Solution:

(i) A+B =

L.H.S. = (A+B)' = 

R.H.S. = A'+B' = 

∴L.H.S = R.H.S.

Hence, proved.

(ii) A-B = 

L.H.S. = (A-B)'

R.H.S. = A'-B' =

∴ L.H.S. = R.H.S.

Hence, proved.

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

(i) (A+B)'=A'+B'

(ii) (A-B)'=A'-B'

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.

Hence, proved.

(ii) A-B = 

∴ L.H.S. =  (A-B)'=

R.H.S.= A'-B' = 

∴ L.H.S. = R.H.S.

Hence, proved.

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

Solution:

Given: A' =and B =

then (A')' =A=

Now, A+2B = 

∴(A+2B)' = 

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

(i) A = and B = 

(ii) A = and B =

Solution:

(i) AB = =

∴  L.H.S. = (AB)′ =

R.H.S.= B′A' = 

∴ L.H.S. = R.H.S.

Hence, proved.

(ii) AB =

∴  L.H.S. = (AB)′ =

Now, R.H.S.=B'A' = 

∴ L.H.S. = R.H.S.

Hence, proved.

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

(ii) A =  ,then verify that A′ A = I.

Solution:

(i)

= I = R.H.S.

∴ L.H.S. = R.H.S.

(ii)

= I = R.H.S.

∴ L.H.S. = R.H.S.

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

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

(i) Given: A =

Now, A'=

∵ A = A'

∴ A is a symmetric matrix.

(ii) Given: A = 

Now, A'=

∵ A = A'

∴ A is a symmetric matrix.

Question 8.  For the 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') = 

Now, B' = (A+A')' = 

∵ B = B'

∴ B=(A+A') is a symmetric matrix.

(ii) Given: A =

Let B = (A-A') =

Now, B' = (A-A')' =

∵ -B = B'

∴ B=(A-A') is a skew symmetric matrix.

Question 9. Find 1/2(A+A') and 1/2(A-A') ,when A =.

Solution:

Given: A = 

∴  A' = 

Now,  A+A' = +

Now, A-A' =

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

(i) 

(ii) 

(iii) 

(iv) 

Solution:

(i) Given : A =

⇒ A'=

Let P = 

and Q = 

Now, P =.....(1)

& P' = 

∵ P=P'

∴ P is a symmetric matrix.

Now, Q =.....(2)

& Q' = 

∵ -Q=Q' 

∴ Q is a skew symmetric matrix.

By adding (1) and (2), we get,

Therefore, A =P + Q

(ii) Given : 

⇒ A'=

P = 

.....(1)

Q = 

......(2)

By adding (1) and (2), we get,

\begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}

Therefore, A =P + Q

(iii) Given: A =

⇒ A'=

P = }.....(1)

Q = ......(2)

By adding (1) and (2), we get

}

Therefore, A =P + Q

(iv) Given: A = 

⇒ A'= 

P =

.....(1)

Q = 

.....(2)

By adding (1) and (2), we get

Therefore, A =P + Q

Question 11. If A, B are symmetric matrices of same order, then 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'

⇒ B=B'

Now, ( AB – BA)' =(AB)'-(BA)'              [∵ (X-Y)'=X'-Y']

                          =B'A'-A'B'                [∵ (XY)'=Y'X']

                         =BA-AB                   [∵ Given]

                        = -(AB-BA)

∴(AB-BA) is a skew symmetric matrix.

∴ The option (A) is correct.

Question 12. If A =, and A + A′ = I, then the value of α is

(A)π/6    (B) π/3

(C) π    (D)3π/2

Solution:

On comparing both sides, we get

           2cosα = 1

⇒      cosα = 

⇒      cosα = cos

⇒      α = 

∴ The option (B) is correct.

Summary

Chapter 3 of the Class 12 NCERT Mathematics textbook, "Matrices," explores essential concepts such as matrix operations and their applications. Exercise 3.3 focuses on practical problems involving matrix addition, subtraction, and multiplication. This exercise provides step-by-step solutions to help students understand and apply matrix operations effectively. Key topics include matrix addition and subtraction, matrix multiplication, determinants, and finding matrix inverses.

• Class 12 NCERT Solutions - Mathematics Part I - Chapter 3 Matrices - Exercise 3.2
• Class 12 NCERT Solutions - Mathematics Part I - Chapter 3 Matrices - Exercise 3.4