NCERT Exemplar Solutions
Class 12 - Mathematics - Chapter 3: Matrices

Short Answer (S.A.)

If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

In the matrix

\( A = \begin{bmatrix} a & 1 & x \\ 2 & \sqrt{3} & x^2 - y \\ 0 & 5 & -\dfrac{2}{5} \end{bmatrix} \), write:

(i) the order of the matrix A, (ii) the number of elements, (iii) the elements \( a_{23}, a_{31}, a_{12} \).

Construct a 2 × 2 matrix where (i) \( a_{ij} = \dfrac{(i - 2j)^2}{2} \) and (ii) \( a_{ij} = | -2i + 3j | \).

Construct a 3 × 2 matrix whose elements are given by \( a_{ij} = e^{ix} \sin(jx) \).

Find the values of \( a \) and \( b \) if matrices A and B are equal.

If possible, find the sum of the matrices A and B.

If \( X = \begin{bmatrix} 3 & 1 & -1 \\ 5 & -2 & -3 \end{bmatrix} \) and \( Y = \begin{bmatrix} 2 & 1 & -1 \\ 7 & 2 & 4 \end{bmatrix} \), find (i) \( X + Y \), (ii) \( 2X - 3Y \), (iii) a matrix \( Z \) such that \( X + Y + Z = 0 \).

Find the non-zero value of \( x \) satisfying the matrix equation.

Show that \( (A + B)(A - B) \neq A^2 - B^2 \).

Find the value of \( x \) if

\( \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = O \).

Show that \( A = \begin{bmatrix} 5 & 3 \\ -1 & -2 \end{bmatrix} \) satisfies the equation \( A^2 - 3A - 7I = O \) and hence find \( A^{-1} \).

Find the matrix \( A \) satisfying the matrix equation:

\( \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \).

Find \( A \) if

\( \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix} \).

If \( A = \begin{bmatrix} 3 & -4 \\ 1 & 1 \\ 2 & 0 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix} \), verify that \( (BA)' \neq B'A^2 \).

If possible, find \( BA \) and \( AB \), where

\( A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix},\ B = \begin{bmatrix} 4 \\ 2 \\ 1 \end{bmatrix} \).

Show by an example that for \( A \neq O \), \( B \neq O \), \( AB = O \).

Given \( A = \begin{bmatrix} 2 & 4 & 0 \\ 3 & 9 & 6 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 4 \\ 2 & 8 \\ 1 & 3 \end{bmatrix} \), is \( (AB)' = B'A' \)?

Solve for \( x \) and \( y \):

\( x \begin{bmatrix} 2 \\ 1 \end{bmatrix} + y \begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatrix} = O \).

If \( X \) and \( Y \) are \( 2 \times 2 \) matrices, solve the matrix equations:

\( 2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix},\ 3X + 2Y = \begin{bmatrix} -2 & 2 \\ 1 & -5 \end{bmatrix} \).

If \( A = [3\ 5] \) and \( B = [7\ 3] \), then find a non-zero matrix \( C \) such that \( AC = BC \).

Give an example of matrices \( A, B, C \) such that \( AB = AC \), where \( A \) is non-zero, but \( B \neq C \).

If \( A = \begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix}, B = \begin{bmatrix} 2 & 3 \\ 3 & -4 \end{bmatrix}, C = \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \), verify (i) \( (AB)C = A(BC) \), (ii) \( A(B + C) = AB + AC \).

If \( P = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \) and \( Q = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \), prove that \( PQ = QP \).

If \( \begin{bmatrix} 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} = A \), find \( A \).

If \( A = \begin{bmatrix} 2 & 1 \end{bmatrix}, B = \begin{bmatrix} 5 & 3 & 4 \\ 8 & 7 & 6 \end{bmatrix}, C = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 0 & 2 \end{bmatrix} \), verify that \( A(B + C) = AB + AC \).

If \( A = \begin{bmatrix} 1 & 0 & -1 \\ 2 & 1 & 3 \\ 0 & 1 & 1 \end{bmatrix} \), then verify that \( A^2 + A = A(A + I) \), where \( I \) is the \( 3 \times 3 \) unit matrix.

If \( A = \begin{bmatrix} 0 & -1 & 2 \\ 4 & 3 & -4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 0 \\ 1 & 3 \\ 2 & 6 \end{bmatrix} \), verify that:

(i) \( (A')' = A \)

(ii) \( (AB)' = B'A' \)

(iii) \( (kA)' = (kA') \)

If \( A = \begin{bmatrix} 1 & 2 \\ 4 & 1 \\ 5 & 6 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 2 \\ 6 & 4 \\ 7 & 3 \end{bmatrix} \), then verify that:

(i) \( (2A + B)' = 2A' + B' \)

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

Show that \( A'A \) and \( AA' \) are both symmetric matrices for any matrix \( A \).

Let \( A \) and \( B \) be square matrices of order \( 3 \times 3 \). Is \( (AB)' = A'B' \)? Give reasons.

Show that if \( A \) and \( B \) are square matrices such that \( AB = BA \), then \( (A + B)^2 = A^2 + 2AB + B^2 \).

Let \( A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix} \), \( B = \begin{bmatrix} 4 & 0 \\ 1 & 5 \end{bmatrix} \), \( C = \begin{bmatrix} 2 & 0 \\ 1 & -2 \end{bmatrix} \) and \( a = 4, b = -2 \). Show that:

(a) \( A + (B + C) = (A + B) + C \)

(b) \( A(BC) = (AB)C \)

(c) \( (a + b)B = aB + bB \)

(d) \( a(C - A) = aC - aA \)

(e) \( (A')' = A \)

(f) \( (bA)' = bA' \)

(g) \( (AB)' = B'A' \)

(h) \( (A - B)C = AC - BC \)

(i) \( (A - B)' = A' - B' \)

If \( A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), then show that \( A^2 = \begin{bmatrix} \cos 2\theta & \sin 2\theta \\ -\sin 2\theta & \cos 2\theta \end{bmatrix} \).

If \( A = \begin{bmatrix} 0 & -x \\ x & 0 \end{bmatrix} \), \( B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) and \( x^2 = -1 \), then show that \( (A + B)^2 = A^2 + B^2 \).

Verify that \( A^2 = I \) when \( A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix} \).

Prove by Mathematical Induction that \( (A^n)' = (A')^n \), where \( n \in \mathbb{N} \) for any square matrix \( A \).

Find inverse, by elementary row operations (if possible), of the following matrices:

(i) \( \begin{bmatrix} 1 & 3 \\ -5 & 7 \end{bmatrix} \)

(ii) \( \begin{bmatrix} 1 & -3 \\ -2 & 6 \end{bmatrix} \)

If \( \begin{bmatrix} xy & 4 \\ z + 6 & x + y \end{bmatrix} = \begin{bmatrix} 8 & w \\ 0 & 6 \end{bmatrix} \), find values of \( x, y, z, w \).

If \( A = \begin{bmatrix} 1 & 5 \\ 7 & 12 \end{bmatrix} \) and \( B = \begin{bmatrix} 9 & 1 \\ 7 & 8 \end{bmatrix} \), find a matrix \( C \) such that \( 3A + 5B + 2C \) is a null matrix.

If \( A = \begin{bmatrix} 3 & -5 \\ -4 & 2 \end{bmatrix} \), then find \( A^2 - 5A - 14I \). Hence, obtain \( A^3 \).

Find the values of \( a, b, c, d \) if

\( 3 \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & 6 \\ -1 & 2d \end{bmatrix} + \begin{bmatrix} 4 & a + b \\ c + d & 3 \end{bmatrix} \).

Find the matrix \( A \) such that

\( \begin{bmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{bmatrix} A = \begin{bmatrix} -1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15 \end{bmatrix} \).

If \( A = \begin{bmatrix} 1 & 2 \\ 4 & 1 \end{bmatrix} \), find \( A^2 + 2A + 7I \).

If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \) and \( A^{-1} = A' \), find value of \( \alpha \).

If the matrix \( \begin{bmatrix} 0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0 \end{bmatrix} \) is a skew-symmetric matrix, find the values of \( a, b, c \).

If \( P(x) = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix} \), then show that

\( P(x) P(y) = P(x + y) = P(y) P(x) \).

If \( A \) is a square matrix such that \( A^2 = A \), show that \( (I + A)^3 = 7A + I \).

If \( A, B \) are square matrices of same order and \( B \) is skew-symmetric, show that \( A' B A \) is skew-symmetric.