NCERT Exemplar Solutions
Class 12 - Mathematics

Chapter 3: Matrices

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.

If \( AB = BA \) for any two square matrices, prove by mathematical induction that \( (AB)^n = A^n B^n \).

Find \( x, y, z \) if

\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]

satisfies \( A' = A^{-1} \), where \( A' \) is the transpose of \( A \).

If possible, using elementary row transformations, find the inverse of the following matrices:

(i) \( A_1 = \begin{bmatrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{bmatrix} \)

(ii) \( A_2 = \begin{bmatrix} 2 & 3 & -3 \\ -1 & -2 & 2 \\ 1 & 1 & -1 \end{bmatrix} \)

(iii) \( A_3 = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \).

Express the matrix

\[ A = \begin{bmatrix} 2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2 \end{bmatrix} \]

as the sum of a symmetric and a skew-symmetric matrix.

_____ matrix is both symmetric and skew symmetric matrix.

Sum of two skew symmetric matrices is always _____ matrix.

The negative of a matrix is obtained by multiplying it by _____.

The product of any matrix by the scalar _____ is the null matrix.

A matrix which is not a square matrix is called a _____ matrix.

Matrix multiplication is _____ over addition.

If A is a symmetric matrix, then A\(^3\) is a _____ matrix.

If A is a skew symmetric matrix, then A\(^2\) is a ____.

If A is skew symmetric, then kA is a _____ (k is any scalar).

If A is symmetric matrix, then B'AB is ____.

If A and B are symmetric matrices of same order, then AB is symmetric if and only if ____.

In applying one or more row operations while finding A\(^{-1}\) by elementary row operations, we obtain all zeros in one or more, then A\(^{-1}\) ____.

If A and B are square matrices of the same order, then (i) (AB)' = ____.

(ii) (kA)' = ____ (k is any scalar)

(iii) [k(A − B)]' = ____.

If A and B are symmetric matrices, then (i) AB − BA is a ____.

(ii) BA − 2AB is a ____.

A matrix denotes a number.

Matrices of any order can be added.

Two matrices are equal if they have same number of rows and same number of columns.

Matrices of different order can not be subtracted.

Matrix addition is associative as well as commutative.

Matrix multiplication is commutative.

A square matrix where every element is unity is called an identity matrix.

If A and B are two square matrices of the same order, then A + B = B + A.

If A and B are two matrices of the same order, then A − B = B − A.

If matrix AB = O, then A = O or B = O or both A and B are null matrices.

Transpose of a column matrix is a column matrix.

If A and B are two square matrices of the same order, then AB = BA.

If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

If A and B are any two matrices of the same order, then (AB)' = A'B'.

If (AB)' = B'A', where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.

If A, B and C are square matrices of same order, then AB = AC always implies that B = C.

AA' is always a symmetric matrix for any matrix A.

If A = \( \begin{bmatrix} 2 & 3 & -1 \\ 1 & 4 & 2 \end{bmatrix} \) and B = \( \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix} \), then AB and BA are defined and equal.

If A is skew symmetric matrix, then A\(^2\) is a symmetric matrix.

(AB)^-1 = A^-1 B^-1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.