RBSE Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 3 Matrices Ex 3.3 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 3 Matrices Ex 3.3

Question 1.
Find the response of each of the following matrices:
(i) \(\left[\begin{array}{r} 5 \\ \frac{1}{2} \\ -1 \end{array}\right]\)
Answer:
\(\left[\begin{array}{lll} 5 & \frac{1}{2} & -1 \end{array}\right]\)

(ii) \(\left[\begin{array}{rr} 1 & -1 \\ 2 & 3 \end{array}\right]\)
Answer:
\(\left[\begin{array}{rr} 1 & 2 \\ -1 & 3 \end{array}\right]\)

(iii) \(\left[\begin{array}{rrr} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{array}\right]\)
Answer:
\(\left[\begin{array}{rrr} -1 & \sqrt{3} & 2 \\ 5 & 5 & 3 \\ 6 & 6 & -1 \end{array}\right]\)

Question 2.
If A = \(\left[\begin{array}{rrr} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{array}\right]\) and B = \(\left[\begin{array}{rrr} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{array}\right]\) then verify that
(i) (A + B)' = A' + B'
Answer:

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

Question 3.
If A' = \(\left[\begin{array}{rr} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{array}\right]\) and B = \(\left[\begin{array}{rrr} -1 & 2 & 1 \\ 1 & 2 & 3 \end{array}\right]\) then verify that
(i) (A + B)' = A' + B'
Answer:

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

Question 4.
If A' = \(\left[\begin{array}{rr} -2 & 3 \\ 1 & 2 \end{array}\right]\) and B = \(\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right]\), then find (A + 2B).
Answer:

Question 5.
For the matrices A and B verify that (AB)' = B'A', where
(i) A = \(\left[\begin{array}{r} 1 \\ -4 \\ 3 \end{array}\right]\)
Answer:

(ii) A = \(\left[\begin{array}{l} 0 \\ 1 \\ 2 \end{array}\right]\), B = [1 5 7]
Answer:

Question 6.
(i) If A = \(\left[\begin{array}{rr} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]\), then verify that AA' = I.
Answer:

(ii) If A = \(\left[\begin{array}{rr} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{array}\right]\) , then verify that A'A = I.
Answer:

Question 7.
(i) Show that the matrix A = \(\left[\begin{array}{rrr} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{array}\right]\) is a symmetric matrix.
Answer:

(ii) Prove that the matrix A = \(\left[\begin{array}{rrr} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{array}\right]\) is a skew-symmetric matrix.
Answer:

Question 8.
For matrix A = \(\left[\begin{array}{ll} 1 & 5 \\ 6 & 7 \end{array}\right]\), verify that
(i) (A + A') is a symmetric matrix
Answer:

= A + A'
∴ (A + A')' = A + A'
Thus, A + A' is a symmetric matrix.

(ii) (A - A') is a skew-symmetric matrix
Answer:

= - (A - A')
∵ (A - A')' = - (A - A')
Thus, A - A' is a skew-symmetric matrix.

Question 9.
If A = \(\left[\begin{array}{rrr} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{array}\right]\), then find \(\frac{1}{2}\) (A + A') and \(\frac{1}{2}\) (A - A').
Answer:

Question 10.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix.
(i) \(\left[\begin{array}{rr} 3 & 5 \\ 1 & -1 \end{array}\right]\)
Answer:
Let A = \(\left[\begin{array}{rr} 3 & 5 \\ 1 & -1 \end{array}\right]\)
We know that any square matrix can be expressed as sum of symmetric and skew-symmetric matrices.
Here, A = \(\left[\begin{array}{rr} 3 & 5 \\ 1 & -1 \end{array}\right]\) then \(\frac{1}{2}\) (A + A') will be symmetric and \(\frac{1}{2}\) (A - A') will be skew-symmetric matrix.

(ii) \(\left[\begin{array}{rrr} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{array}\right]\)
Answer:

(iii) \(\left[\begin{array}{rrr} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{array}\right]\)
Answer:

(iv) \(\left[\begin{array}{rr} 1 & 5 \\ -1 & 2 \end{array}\right]\)
Answer:

Choose the correct answer in the exercises 11 and 12.

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
Answer:
Matrix A and B are symmetric matrix of equal order.
∴ A' = A, B' = B
(AB - BA)' = (AB)' - (BA)'

= - (AB - BA)
= skew-symmetric matrix
= (AB - BA) skew-symmetric matrix.
Thus, (A) is correct.

Question 12.
If A = \(\left[\begin{array}{rr} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]\), then A + A' = I, then the value of α is:
(A) \(\frac{\pi}{6}\)
(B) \(\frac{\pi}{3}\)
(C) π
(D) \(\frac{3 \pi}{2}\)
Answer: