Matrices: 1 mark Questions (CBSE Class 12)

Which of the following cannot be the order of a row-matrix ?

1. (A)$2\times 1$
2. (B)$1\times 2$
3. (C)$1\times 1$
4. (D)$1\times n$

Which of the following properties is/are true for two matrices of suitable orders ?
(i) $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }$
(ii) $(A-B{)}^{\prime }=B^{\prime }-A^{\prime }$
(iii) $(AB{)}^{\prime }=A^{\prime }{B}^{\prime }$
(iv) $(kAB{)}^{\prime }=k{B}^{\prime }{A}^{\prime }$ (k is a scalar)

1. (A)(i) only
2. (B)(i), (ii) and (iii)
3. (C)(i) and (ii)
4. (D)(i) and (iv)

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric ?

1. (A)AB
2. (B)AB + BA
3. (C)$(A+B{)}^2$
4. (D)$A-B$

Which of the following cannot be an order of a column-matrix ?

1. (A)$1\times 2$
2. (B)$2\times 1$
3. (C)$1\times 1$
4. (D)$m\times 1$, where $m\in N$

If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?

1. (A)$A^2+B^2$
2. (B)$A^2-B^2$
3. (C)AB + BA
4. (D)$AB-BA$

If $A=[a_{ij}{]}_{3\times 3}$ is a scalar matrix then which of the following must be true ?

1. (A)A must be a symmetric matrix.
2. (B)A must be a skew-symmetric matrix.
3. (C)A must be an identity matrix.
4. (D)A must be a null matrix.

If A and B are symmetric matrics of same order, then (AB – BA) is a

1. (A)Zero matrix
2. (B)Identity matrix
3. (C)Symmetric matrix
4. (D)Skew symmetric matrix

If A and B are square matrices of same order, then which of the following statements is/are always true ?
(i) $(A+B)(A-B)=A^2-B^2$
(ii) $AB=BA$
(iii) $(A+B{)}^2=A^2+AB+BA+B^2$
(iv) $AB=0\Rightarrow A=0$ or $B=0$

1. (A)Only (i) and (iii)
2. (B)Only (ii) and (iii)
3. (C)Only (iii)
4. (D)Only (iii) and (iv)

If $A=\left[\begin{array}{ccc}1& a& b\\ -1& 2& c\\ 0& 5& 3\end{array}\right]$ is a symmetric matrix, then the value of $3a+b+c$ is

1. (A)2
2. (B)6
3. (C)4
4. (D)0

If $A=\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right]$ and $A+A^{\prime }=I$, then the value of $x\in \left[0,\frac{\pi }{2}\right]$ is

1. (A)$0$
2. (B)$\frac{\pi }{4}$
3. (C)$\frac{\pi }{3}$
4. (D)$\frac{\pi }{2}$

For a square matrix A, $(3A{)}^{-1}=$

1. (A)$3A^{-1}$
2. (B)$9A^{-1}$
3. (C)$\frac{1}{3}{A}^{-1}$
4. (D)$\frac{1}{9}{A}^{-1}$

If $A=\left[\begin{array}{cc}\frac{1}{2}\cos x& -\sin x\\ \sin x& \frac{1}{2}\cos x\end{array}\right]$ and $A+A^{\prime }=I$, then value of $x\in \left[\frac{-\pi }{2},\frac{\pi }{2}\right]$ is

1. (A)$\frac{\pi }{2}$
2. (B)$\frac{\pi }{3}$
3. (C)$0$
4. (D)$\frac{-\pi }{2}$

If $A=\left[\begin{array}{cc}\tan x& \cot x\\ -\cot x& \tan x\end{array}\right]$ and $A+A^{\prime }=2I$, then value of $x\in \left[0,\frac{\pi }{2}\right]$ is

1. (A)$0$
2. (B)$\frac{\pi }{3}$
3. (C)$\frac{\pi }{4}$
4. (D)$\frac{\pi }{2}$

If $A^2=4A+3I$ and $A^{-1}=xA+yI$, then the value of $(x+y)$ is :

1. (A)$-1$
2. (B)$1$
3. (C)$\frac{5}{3}$
4. (D)$7$

If A and B are skew-symmetric matrices of same order, then $A{B}^{\prime }+B{A}^{\prime }$ is a/an :

1. (A)symmetric matrix
2. (B)skew-symmetric matrix
3. (C)null matrix
4. (D)identity matrix

If $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]A=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$, then order of A must be :

1. (A)$3\times 1$
2. (B)$1\times 3$
3. (C)$1\times 1$
4. (D)$3\times 3$

If a square matrix A is such that $A^2=A$ and $(I-A{)}^3=xA+I$, then value of x must be :

1. (A)$7$
2. (B)$5$
3. (C)$-7$
4. (D)$-1$

If a matrix B is such that $B[123]=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ 2& 0& 1\end{array}\right]$, then the order of matrix B is :

1. (A)$1\times 3$
2. (B)$3\times 1$
3. (C)$3\times 3$
4. (D)$1\times 1$

If a matrix X is such that $[21]X=[345]$, then the order of matrix X is :

1. (A)$3\times 1$
2. (B)$2\times 3$
3. (C)$1\times 2$
4. (D)$1\times 3$

If $A=[a_{ij}]$ is a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{|i-3j|}{2}$, then $A^{\prime }$ is :

1. (A)$\left[\begin{array}{cc}1& \frac{5}{2}\\ \frac{1}{2}& 2\end{array}\right]$
2. (B)$\left[\begin{array}{cc}1& \frac{1}{2}\\ \frac{5}{2}& 2\end{array}\right]$
3. (C)$\left[\begin{array}{cc}2& \frac{5}{2}\\ \frac{1}{2}& 1\end{array}\right]$
4. (D)$\left[\begin{array}{cc}2& \frac{1}{2}\\ \frac{5}{2}& 1\end{array}\right]$

Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Let $A=[a_{ij}]$ be a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{(2i-j{)}^2}{3}$. Then $A^{\prime }$ is :

1. (A)$\left[\begin{array}{cc}\frac{1}{3}& 3\\ 0& \frac{4}{3}\end{array}\right]$
2. (B)$\left[\begin{array}{cc}\frac{1}{3}& 0\\ 3& \frac{4}{3}\end{array}\right]$
3. (C)$\left[\begin{array}{cc}\frac{4}{3}& 3\\ 0& \frac{1}{3}\end{array}\right]$
4. (D)$\left[\begin{array}{cc}\frac{4}{3}& 0\\ 3& \frac{1}{3}\end{array}\right]$

Let $A=[a_{ij}]$ be a $2\times 2$ matrix whose elements are given by $a_{ij}=\frac{(i+2j{)}^2}{3}$. Then $A^{\prime }$ is given by :

1. (A)$\left[\begin{array}{cc}12& \frac{25}{3}\\ \frac{16}{3}& 3\end{array}\right]$
2. (B)$\left[\begin{array}{cc}12& \frac{16}{3}\\ \frac{25}{3}& 3\end{array}\right]$
3. (C)$\left[\begin{array}{cc}3& \frac{25}{3}\\ \frac{16}{3}& 12\end{array}\right]$
4. (D)$\left[\begin{array}{cc}3& \frac{16}{3}\\ \frac{25}{3}& 12\end{array}\right]$

If matrix $A=\left[\begin{array}{cc}-p& q\\ r& p\end{array}\right]$ is such that $A^2=I$, then :

1. (A)$1+p^2+qr=0$
2. (B)$1-p^2-qr=0$
3. (C)$1-p^2+qr=0$
4. (D)$1+p^2-qr=0$

If A is a square matrix such that $A^2=A$, then $(A-I{)}^3-A$ is equal to :

1. (A)$I$
2. (B)$-I$
3. (C)$A$
4. (D)$A^2$

Let $A=\left[\begin{array}{ccc}0& -3& 4\\ 1& 0& 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}-3& 0& 1\\ 2& 4& 0\end{array}\right]$. If A + B + C = O, then matrix C is :

1. (A)$\left[\begin{array}{ccc}-3& -3& 5\\ 3& 4& 2\end{array}\right]$
2. (B)$\left[\begin{array}{ccc}3& 3& 5\\ -3& -4& -2\end{array}\right]$
3. (C)$\left[\begin{array}{ccc}3& 3& -5\\ -3& -4& -2\end{array}\right]$
4. (D)$\left[\begin{array}{ccc}-3& -3& -5\\ 3& 4& 2\end{array}\right]$

If $A=\left[\begin{array}{cc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{array}\right]$ and $A+A^{\prime }=I$, then $\theta$ is equal to :

1. (A)0
2. (B)$\frac{\pi }{6}$
3. (C)$\frac{\pi }{3}$
4. (D)$\pi$

If A is a symmetric matrix, then for any matrix B of order same as A, $BA{B}^{\prime }$ is a/an :

1. (A)Skew symmetric matrix
2. (B)Identity matrix
3. (C)Symmetric matrix
4. (D)Null matrix

For any square matrix A with real entries, if $A+A^{\prime }$ is a symmetric matrix then :

1. (A)$(A-A^{\prime })$ cannot be a skew symmetric matrix
2. (B)$(A-A^{\prime })$ is a skew symmetric matrix
3. (C)A is always a symmetric matrix
4. (D)A is always a skew symmetric matrix

A matrix $B=[b_{ij}{]}_{m\times m}$ is said to be a diagonal matrix, if :

1. (A)$b_{ij}=0$, when $i=j$
2. (B)$b_{ij}=1$, when $i=j$
3. (C)$b_{ij}=1$, when $i\ne j$
4. (D)$b_{ij}=0$, when $i\ne j$