CBSE Class 12 Mathematics Matrices Important Questions Set 01

Question. Fill in the blanks.
(i) If \( A \) and \( B \) are symmetric matrices of same order then \( AB \) is symmetric if and only if \( AB = BA \).
(ii) If \( \begin{bmatrix} x+y & 7 \\ 9 & x-y \end{bmatrix} = \begin{bmatrix} 2 & 7 \\ 9 & 4 \end{bmatrix} \), then \( x.y = -3 \). 
(iii) If \( \begin{bmatrix} x & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 4 \end{bmatrix} = 0 \), then \( x = -8 \).
(iv) If \( A \) is symmetric matrix, then \( B' AB \) is symmetric.

Question. For a \( 2 \times 2 \) matrix, \( A = [a_{ij}] \), whose elements are given by \( a_{ij} = \frac{i}{j} \), write the value of \( a_{12} \). 
Answer: \( a_{12} = \frac{1}{2} \)

Question. Write the order of the product matrix. 
\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 \end{bmatrix} \)
Answer: \( 3 \times 3 \)

Question. From the following matrix equation, find the value of \( x \) :
\( \begin{bmatrix} x + y & 4 \\ -5 & 3y \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ -5 & 6 \end{bmatrix} \) 
Answer: \( x = 1 \)

Question. If \( \begin{bmatrix} 3x - 2y & 5 \\ x & -2 \end{bmatrix} = \begin{bmatrix} 3 & 5 \\ -3 & -2 \end{bmatrix} \), then find the value of \( y \). 
Answer: \( -6 \)

Question. Write a square matrix of order 2, which is both symmetric and skew symmetric. 
Answer: \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Question. If matrix \( A = [1\ 2\ 3] \), then write \( AA' \), where \( A' \) is the transpose of matrix \( A \). 
Answer: \( [14] \)

Question. If the matrix \( A = \begin{bmatrix} 0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0 \end{bmatrix} \) is skew symmetric, find the values of ‘a’ and ‘b’. 
Answer: \( a = -2, b = 3 \)

Question. If \( A \) is a square matrix such that \( A^2 = A \), then write the value of \( (I + A)^3 - 7A \)
Answer: \( I \)

Question. If a matrix has 5 elements, write all possible orders it can have. 
Answer: \( 1 \times 5 \) and \( 5 \times 1 \)

Question. Write the element \( a_{23} \) of a \( 3 \times 3 \) matrix \( A = (a_{ij}) \) whose elements \( a_{ij} \) are given by \( a_{ij} = \frac{|i - j|}{2} \). 
Answer: \( \frac{1}{2} \)

Question. In the matrix equation \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 2 & -1 \end{pmatrix} = \begin{pmatrix} 8 & -3 \\ 9 & -4 \end{pmatrix} \). Use elementary operation \( R_2 \rightarrow R_2 + R_1 \) and write the equation thus obtained.
Answer: \( \begin{bmatrix} 2 & 3 \\ 3 & 7 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix} = \begin{bmatrix} 8 & -3 \\ 17 & -7 \end{bmatrix} \)

Question. Write the number of all possible matrices of order \( 2 \times 2 \) with each entry 1, 2 or 3. 
Answer: \( 81 \)

Short Answer Questions-I: 

Question. Find a matrix \( A \) such that \( 2A - 3B + 5C = O \), where \( B = \begin{bmatrix} -2 & 2 & 0 \\ 3 & 1 & 4 \end{bmatrix} \) and \( C = \begin{bmatrix} 2 & 0 & -2 \\ 7 & 1 & 6 \end{bmatrix} \). 
Answer: \( \begin{bmatrix} -8 & 3 & 5 \\ -13 & -1 & -9 \end{bmatrix} \)

Question. If \( A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix} \) and \( kA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix} \), then find the value of \( k, a \) and \( b \). 
Answer: \( k = -6, a = -4, b = -9 \)

Question. Express \( A = \begin{bmatrix} 4 & -3 \\ 2 & -1 \end{bmatrix} \) as a sum of a symmetric and a skew-symmetric matrix. 
Answer: \( \begin{bmatrix} 4 & -1/2 \\ -1/2 & -1 \end{bmatrix} + \begin{bmatrix} 0 & -5/2 \\ 5/2 & 0 \end{bmatrix} \)

Question. Solve the following matrix equation for \( x \): \( \begin{bmatrix} x & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = O \).
Answer: \( x = 2 \)

Question. If \( 2 \begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix} \), find \( (x - y) \). 
Answer: \( 10 \)

Question. If \( A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix} \) and \( B = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix} \), then find \( (3A - B) \). 
Answer: \( \begin{bmatrix} 8 & 7 \\ 6 & 2 \end{bmatrix} \)

Question. If \( \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \begin{bmatrix} 1 & -3 \\ -2 & 4 \end{bmatrix} = \begin{bmatrix} -4 & 6 \\ -9 & x \end{bmatrix} \), then write the value of \( x \). 
Answer: \( x = 13 \)

Question. If matrix \( A = \begin{bmatrix} 3 & -3 \\ -3 & 3 \end{bmatrix} \) and \( A^2 = \lambda A \), then write the value of \( \lambda \)
Answer: \( \lambda = 6 \)

Question. If matrix \( A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \) and \( A^2 = pA \), then write the value of \( p \). 
Answer: \( p = 4 \)

Short Answer Questions-II:

Question. Given matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), find \( f(A) \), if \( f(x) = 2x^2 - 3x + 5 \).
Answer: \( \begin{bmatrix} 16 & 14 \\ 21 & 37 \end{bmatrix} \)

Question. Find the matrix \( X \) such that \( \begin{bmatrix} 2 & -1 \\ 0 & 1 \\ -2 & 4 \end{bmatrix} X = \begin{bmatrix} -1 & -8 & -10 \\ 3 & 4 & 0 \\ 10 & 20 & 10 \end{bmatrix} \).
Answer: \( X = \begin{bmatrix} 1 & -2 & -5 \\ 3 & 4 & 0 \end{bmatrix} \)

Question. Express the matrix \( \begin{bmatrix} 2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2 \end{bmatrix} \) as the sum of a symmetric and a skew symmetric matrix.
Answer: \( \begin{bmatrix} 2 & 2 & 5/2 \\ 2 & -1 & 3/2 \\ 5/2 & 3/2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 1 & -3/2 \\ -1 & 0 & 1/2 \\ 3/2 & -1/2 & 0 \end{bmatrix} \)

Question. If \( A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \), then find the value of \( A^2 - 3A + 2I \).
Answer: \( \begin{bmatrix} 1 & -1 & -1 \\ 3 & -3 & -4 \\ -3 & 2 & 0 \end{bmatrix} \)

Question. Show that the elements along the main diagonal of a skew symmetric matrix are all zero. 
Answer: For a skew-symmetric matrix \( A = [a_{ij}] \), we have \( a_{ij} = -a_{ji} \) for all \( i, j \). For diagonal elements \( i = j \), therefore \( a_{ii} = -a_{ii} \), which implies \( 2a_{ii} = 0 \), hence \( a_{ii} = 0 \) for all \( i \).

Question. If \( A = \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{bmatrix}, C = \begin{bmatrix} 2 \\ -2 \\ 3 \end{bmatrix} \), then calculate \( AC, BC \) and \( (A + B)C \). Also verify that \( (A + B)C = AC + BC \).
Answer: \( AC = \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix} \), \( BC = \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix} \), \( (A + B)C = \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix} \).
Since \( AC + BC = \begin{bmatrix} 9 \\ 12 \\ 30 \end{bmatrix} + \begin{bmatrix} 1 \\ 8 \\ -2 \end{bmatrix} = \begin{bmatrix} 10 \\ 20 \\ 28 \end{bmatrix} \), verification is complete.

Question. A manufacturer produces three products \( x, y, z \) which he sells in two markets. Annual sales are indicated in the table:
Market I: \( x = 10,000, y = 2,000, z = 18,000 \)
Market II: \( x = 6,000, y = 20,000, z = 8,000 \)
If unit sale price of \( x, y \) and \( z \) are ₹2.50, ₹1.50 and ₹1.00 respectively, then find the total revenue in each market, using matrices.
Answer: Market I : ₹46,000 ; Market II : ₹53,000

Question. Choose and write the correct option in each of the following questions.
(i) If \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} y & 4x \\ 6y & 4 \end{bmatrix} \) then
(a) \( x = 2, y = 2 \)
(b) \( x = \frac{1}{2}, y = \frac{1}{2} \)
(c) \( x = \frac{1}{2}, y = 2 \)
(d) \( x = 2, y = \frac{1}{2} \)
Answer: (b)

Question. If \( A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}, B = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \), where \( i = \sqrt{-1} \), then the correct relation is
(a) \( A + B = 0 \)
(b) \( A^2 = B^2 \)
(c) \( A - B = 0 \)
(d) \( A^2 + B^2 = 0 \)
Answer: (b)

Question. A square matrix \( A = [a_{ij}] \) in which \( a_{ij} = 0 \) for \( i \neq j \) and \( a_{ij} = k \) (Constant) for \( i = j \) is called a
(a) Unit matrix
(b) Scalar matrix
(c) Null matrix
(d) Diagonal matrix
Answer: (b)

Question. For the matrix \( A = \begin{bmatrix} 3 & 1 \\ 7 & 5 \end{bmatrix} \), find \( x \) and \( y \) so that \( A^2 + xI = yA \).
(a) (8, 8)
(b) (–8, 0)
(c) (–8, –8)
(d) None of these
Answer: (a)

Question. Fill in the blanks.
(i) If \( \begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & 3 \\ 1 & 5 \end{bmatrix} \) then \( x = \) _____________ and \( y = \) _____________ .
Answer: 1, 4

Question. If \( A \) and \( B \) are square matrices of the same order, then \( [k(A - B)]' = \) _____________ , where \( k \) is any scalar.
Answer: \( k(A' - B') \)

Question. If \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), then write \( A^n \).
Answer: \( \begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix} \)

Question. If \( \begin{bmatrix} \cos \frac{2\pi}{7} & -\sin \frac{2\pi}{7} \\ \sin \frac{2\pi}{7} & \cos \frac{2\pi}{7} \end{bmatrix}^k = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), then find the least positive integral value of \( k \).
Answer: \( k = 7 \)

Question. If \( \begin{bmatrix} 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix} = A \), then find the value of \( A \).
Answer: \( [-4] \)

Question. Solve for \( x \), \( \begin{bmatrix} 1 & x \end{bmatrix} \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 3 \end{bmatrix} = [0] \).
Answer: \( \frac{1}{7} \)

Question. Find the value of \( x \) and \( y \) which makes the following pair of matrices equal:
\( \begin{bmatrix} 3x + 7 & 5 \\ y + 1 & 2 - 3x \end{bmatrix} = \begin{bmatrix} 0 & y - 2 \\ 8 & 4 \end{bmatrix} \)
Answer: not possible

Question. If \( 2 \begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix} \), find \( (x - y) \).
Answer: 10

Question. If matrix \( A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \) and \( A^2 = pA \), then write the value of \( p \).
Answer: \( p = 4 \)

Question. Find the value of \( x \), if \( \begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = [0] \).
Answer: –14, –2

Question. Show that \( A = \begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix} \) satisfies the equation \( x^2 - 6x + 17 = 0 \). Hence, find \( A^{-1} \).
Answer: \( \frac{1}{17} \begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix} \)

Question. Let \( A = \begin{bmatrix} 3 & 2 & 5 \\ 4 & 1 & 3 \\ 0 & 6 & 7 \end{bmatrix} \), express \( A \) as a sum of two matrices such that one is symmetric and other is skew symmetric.
Answer: \( \begin{bmatrix} 3 & 3 & 5/2 \\ 3 & 1 & 9/2 \\ 5/2 & 9/2 & 7 \end{bmatrix} + \begin{bmatrix} 0 & -1 & 5/2 \\ 1 & 0 & -3/2 \\ -5/2 & 3/2 & 0 \end{bmatrix} \)

Question. (i) Prove that the sum of two skew-symmetric matrices is a skew-symmetric matrix.
(ii) Express the following matrix as the sum of a symmetric and a skew-symmetric matrix.
\( \begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix} \)
Answer: (ii) \( \frac{1}{2} \begin{bmatrix} 2 & -3 & 1 \\ -3 & 16 & 9 \\ 1 & 9 & 10 \end{bmatrix} + \frac{1}{2} \begin{bmatrix} 0 & 9 & 9 \\ -9 & 0 & -3 \\ -9 & 3 & 0 \end{bmatrix} \)