Updated On May 15, 2024

By SaraNextGen

Ch.3 Matrices : Multiple Choice Questions

1) The system of linear equation $a x+b y=0, c x+d y=0$ has non-trivial solution if
(a) $\quad a d-b c=0$
(b) $\quad \mathrm{ad}-\mathrm{bc}<0$
(c) $a d+b c=0$
(d) $\quad a c+b d=0$
Answer: (a)
Explanation: The given system of equations has a non-trivial solution if $\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=0 \Longrightarrow \mathrm{ad}-\mathrm{bc}=0$

2) For what value of $\lambda$ the following system of equations does not have a solution $x+y+z=6$ $4 \mathrm{x}+\lambda \mathrm{y}-\lambda \mathrm{z}=0,3 \mathrm{x}+2 \mathrm{y}-4 \mathrm{z}=-5 ?$
(a) 1
(b) $-3$
(c) 0
(d) 3
Answer:
(d)
Explanation: The given system of equations does not have solution if $\left|\begin{array}{ccc}1 & 1 & 1 \\ 4 & \lambda & -\lambda \\ 3 & 2 & -4\end{array}\right|=0$
$
\begin{aligned}
&\Rightarrow \quad\left|\begin{array}{ccc}
0 & 0 & 1 \\
4+\lambda & 2 \lambda & -\lambda \\
7 & 6 & 4
\end{array}\right|=0 \\
&\Rightarrow(24+6 \lambda-14 \lambda)=0 \Longrightarrow \lambda=3
\end{aligned}
$

3) If $A$ is any square matrix, then
(a) $\mathrm{A}-\mathrm{A}^{\mathrm{t}}$ is symmetric
(b) None of these
(c) $\mathrm{A}+\mathrm{A}^{\mathrm{t}}$ is symmetric
(d) $\quad A+A^{t}$ is skew-symmetric
Answer: (C)
Explanation:
Every square matrix $\left(\mathrm{A}+\mathrm{A}^{\prime}\right)$ is always symmetric.

4) If $\mathrm{I}_{\mathrm{n}}$ is the identity matrix of order $\mathrm{n}$, the $\left(\mathrm{I}_{\mathrm{n}}\right)^{-1}$
(a) Does not exist
(b) $\mathrm{n}_{\mathrm{m}}$
(c) $=\mathrm{I}_{\mathrm{n}}$
(d) $=0$
Answer: (c)
Explanation:
Inverse of any identity matrix is always an identity matrix.

5) The system of equations, $x+2 y=5,4 x+8 y=20$ has
(a) No solution
(b) $\quad$ None of these
(c) A unique solution
(d) Infinitely many solutions
Answer: (d)
Explanation:
For infinitely many solutions, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$,
for given system of equations we have: $\frac{1}{4}=\frac{2}{8}=\frac{5}{20}$

6) If $A$ is a skew-symmetric matrix and $n$ is a positive integer, then $A_{m}^{n}$ is
(a) Symmetric matrix
(b) Skew-symmetric matrix
(c) Diagonal matrix
(d) None of these
Answer: (d)
Explanation:
It is given that $A$ is a skew -symmetric matrix
$
\begin{gathered}
A^{T}=-A \\
\Rightarrow\left(A^{T}\right)^{n}=\left(-A^{\mathrm{n}}\right)^{\mathrm{T}}
\end{gathered}
$
Hence, $\mathrm{A}^{\mathrm{n}} \mid$ is skew-symmetric when ' $\mathrm{n}^{\prime}$ is odd $\& \mathrm{~A}^{\mathrm{n}}$ is symmetric when ' $\mathrm{n}^{\prime}$ is even.

7) If the systems of equations $x+4 a y+a z=0, x+3 b y+b z=0$ and $x+c y+c z=0$ have a non-zero solution then $\mathrm{a}, \mathrm{b}, \mathrm{c}$, are in
(a) G.P
(b) A.P
(c) None of these
(d) H.P
Answer: (d)
Explanation:
For a non-trivial solution: $\left|\begin{array}{lll}1 & 4 a & a \\ 1 & 3 b & b \\ 1 & 2 c & c\end{array}\right|:=0$
$
\begin{aligned}
&\Longrightarrow\left|\begin{array}{ccc}
1 & 4 a & a \\
0 & 3 b-4 a & b-a \\
0 & 2 c-4 a & c-a
\end{array}\right|=0 \\
&\Rightarrow b c+a b-2 a c=0 \\
&\Rightarrow \frac{2}{b}+\frac{1}{a}+\frac{1}{c}
\end{aligned}
$

8) The equations $x+2 y+2 z=1$ and $2 x+4 y+4 z=9$ have
(a) No solution
(b) Only one solution
(c) Only two solutions
(d) Infinitely many solution
Answer: (a)
Explanation: The given system of equations does not have solution if $\left|\begin{array}{ccc}1 & 2 & 2 \\ 2 & 4 & 4\end{array}\right|=0$

9) $\left|\begin{array}{ccc}1 & 1 & 1 \\ e & 0 & \sqrt{2} \\ 2 & 2 & 2\end{array}\right|$ is equal to
(a) 0
(b) $3 \mathrm{e}$
(c) None of these
(d) 2
Answer: (a)
Explanation:
$
\left|\begin{array}{ccc}
1 & 1 & 1 \\
e & 0 & \sqrt{2} \\
2 & 2 & 2
\end{array}\right|=0
$
Because, row 1 and row 3 are proportional.

10) If A and B any two square matrices of the same order, then
(a) $\operatorname{adj}(\mathrm{AB})=\operatorname{adj}(\mathrm{A}) \operatorname{adj}(\mathrm{B})$
(b) $\quad(\mathrm{AB})^{\mathrm{t}}=\mathrm{B}^{\mathrm{t}} \mathrm{A}^{\mathrm{t}}$
(c) $\mathrm{AB}=0$
(d) $\quad(\mathrm{AB})^{\mathrm{t}}=\mathrm{A}^{\mathrm{t}} \mathrm{B}^{\mathrm{t}}$
Answer:
(b)
Explanation:
By the property of transpose of a matrix,
$
A B^{\prime}=B^{\prime} A^{\prime}
$

11) The order of the single matrix obtained from
$$
\left[\begin{array}{cc}
1 & -1 \\
0 & 2 \\
2 & 3
\end{array}\right]_{3 \times 2}\left\{\left[\begin{array}{ccc}
-1 & 0 & 2 \\
2 & 0 & 1
\end{array}\right]_{2 \times 3}-\left[\begin{array}{ccc}
0 & 1 & 23 \\
1 & 0 & 21
\end{array}\right]_{2 \times 3}\right\}
$$
(a) $2 \times 3$
(b) $\quad 3 \times 3$
(c) $3 \times 2$
(d) $\quad 2 \times 2$
Answer: (b)
Explanation:
$
\left[\begin{array}{cc}
1 & -1 \\
0 & 2 \\
2 & 3
\end{array}\right]_{3 \times 2}\left\{\left[\begin{array}{ccc}
-1 & 0 & 2 \\
2 & 0 & 1
\end{array}\right]_{2 \times 3}-\left[\begin{array}{ccc}
0 & 1 & 23 \\
1 & 0 & 21
\end{array}\right]_{2 \times 3}\right\}
$
$
\left[\begin{array}{cc}
1 & -1 \\
0 & 2 \\
2 & 3
\end{array}\right]_{3 \times 2}\left[\begin{array}{ccc}
-1 & -1 & -21 \\
1 & 0 & -20
\end{array}\right]_{2 \times 3}=
$
$
\left[\begin{array}{ccc}
-2 & -1 & -1 \\
2 & 0 & -40 \\
1 & -2 & -102
\end{array}\right]_{3 \times 3}=
$

12) If ' $A^{\prime}$ is square matrix such that $A^{2}=1$, then $A^{-1}$ is equal to
(a) 0
(b) $\quad \mathrm{A}+\mathrm{I}$
(c) I
(d) $\mathrm{A} \mid$
Answer:
(d)
Explanation:
If $A$ and $B$ are two square matrices of the same order and the product $A B=1$, then the matrix $\mathrm{B}$ is called the inverse of matrix $\mathrm{A}$. Therefore, if $A^{2}=1$, then $A$ is the inverse of itself.

13) If $A$ and $B$ are two matrices such that $A+B$ and $A B$ are both defined, then
(a) number of columns of $\mathrm{A}=$ number of rows of $\mathrm{B}$.
(b) $A, B$ are square matrices not necessarily of same order
(c) A and B can be any matrices
(d) A and B are square matrices of same order
Answer: $\quad$ (d)
Explanation:
If $A$ and $B$ are square matrices of the same order, both operations $A+B$ and $A B$ are well defined.

14) The system of equations $3 x+y-z=0,5 x+2 y-3 z=2,15 x+6 y-9 z=5$ has
(a) A unique solution
(b) $\quad$ Two distinct solutions
(c) No solution
(d) Infinitely many solutions
Answer:  (c)
Explanation:
The given system of equations does not has a solution if:
$
\begin{aligned}
&\Rightarrow\left|\begin{array}{ccc}
3 & 1 & -1 \\
5 & 2 & -3 \\
15 & 6 & -9
\end{array}\right|=0 \\
&\Rightarrow 3(-18+18)-1(-45+45)-1(30-30)=0
\end{aligned}
$

15) If $A$ and $B$ are symmetric matrices of order $n(A \neq B)$, then
(a) $\mathrm{A}+\mathrm{B}$ is skew symmetric
(b) $\mathrm{A}+\mathrm{B}$ is a diagonal matrix
(c) $A+B$ is a zero matrix
(d) $\quad A+B$ is symmetric
Answer: $\quad$ (d)
Explanation:
Sum of two symmetric matrices is also symmetric.

16) Each diagonal element of a skew-symmetric matrix is
(a) Negative
(b) $\quad$ Non-real
(c) Positive
(d) Zero
Answer: $\quad$ (d)
Explanation:
The diagonal elements of a skew-symmetric matrix are always zero.

17) The value of $\lambda$, for which system of equations $x+y+z=1, x+2 y+2 z=3, x+2 y+\lambda z=4$, have no solution is
(a) 0
(b) 1
(c) 3
(d) 2
Answer: (d)
Explanation:
The given system of equations does not has a solution if:
$
\begin{aligned}
&\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & 2 \\
1 & 2 & \lambda
\end{array}\right|=0 \\
&\Rightarrow 1(2 \lambda-4)-1(\lambda-2)+1(2-2) \\
&\Rightarrow \lambda=2 .
\end{aligned}
$

18) If $A$ and $B$ are square matrices of the same order and $A B=3 I$, then $A^{-1}$ is equal to
(a) $3 \mathrm{~B}^{-1}$
(b) $3 \mathrm{~B}$
(c) $\frac{1}{3} \mathrm{~B}$
(d) None of these
Answer: $\quad$ (c)
Explanation:
If $A$ and $B$ are square matrices of the same order and $A B=3 I$, then $\frac{1}{3} \mathrm{AB}=\mathrm{I} \mathrm{A}^{-1}=\frac{1}{3} \mathrm{~B}$.

19) If $A$ and $B$ are two matrices such that $A B=B A$ and $B A=A$, then $A^{2}+B^{2}=$
(a) $\mathrm{A}+\mathrm{B}$
(b) $\quad 2 \mathrm{BA}$
(c) $\mathrm{AB}$
(d) $2 \mathrm{AB}$
Answer: (a)
Explanation: $\mathrm{AB}=\mathrm{B}$ $\Rightarrow(\mathrm{AB}) \mathrm{A}=\mathrm{BA}$ $\Rightarrow \mathrm{A}(\mathrm{BA})=\mathrm{BA}$ $\Rightarrow \mathrm{A}(\mathrm{A})=\mathrm{A}$ $\Rightarrow \mathrm{A}^{2}=\mathrm{A}$ $\mathrm{AB}=\mathrm{B} \Rightarrow \mathrm{B}(\mathrm{AB})=\mathrm{BB}$ $\Rightarrow(\mathrm{BA}) \mathrm{B}=\mathrm{B}^{2}$ $\Rightarrow \mathrm{AB}=\mathrm{B}^{2}$ $\Rightarrow \mathrm{B}=\mathrm{B}^{2}$ $\mathrm{~A}^{2}+\mathrm{B}^{2}=\mathrm{A}+\mathrm{B}$
$
\begin{aligned}
&\Rightarrow \mathrm{A}(\mathrm{BA})=\mathrm{BA} \\
&\Rightarrow \mathrm{A}(\mathrm{A})=\mathrm{A}
\end{aligned}
$

20) If $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ then $A^{4}=$
(a) $\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$
(b) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
(c) $\left[\begin{array}{ll}0 & 0 \\ 1 & 1\end{array}\right]$
(d) $\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
Answer: (b)
Explanation:
As the given matrix is an identity matrix, and $\mathrm{I}^{\mathrm{n}}=$ I.I.I $\ldots \mathrm{I}(\mathrm{n}$ times $)=\mathrm{I}$.

21) If for a matrix $\mathrm{A}, \mathrm{A}^{2}+\mathrm{I}=0$ where $\mathrm{I}$ is the identity matrix, then A equals
(a) $\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right]$
(b) $\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$
(c) $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
(d) $\left[\begin{array}{cc}1 & 2 \\ -1 & 1\end{array}\right]$
Answer: (a)
Explanation:
$
\begin{aligned}
&\mathrm{A}^{2}+\mathrm{I}=\mathrm{A} \cdot \mathrm{A}+\mathrm{I} \\
&{\left[\begin{array}{cc}
i & 0 \\
0 & -i
\end{array}\right]\left[\begin{array}{cc}
i & 0 \\
0 & -i
\end{array}\right]+\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]} \\
&\Longrightarrow\left[\begin{array}{cc}
i^{2} & 0 \\
0 & i^{2}
\end{array}\right]+\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] \\
&\Rightarrow\left[\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right]+\left[\begin{array}{cc}
1 & 0 \\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right]=O
\end{aligned}
$

22) If $A+B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$ and A-2B $=\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$, then A=
(a) $\left[\begin{array}{ll}1 & 1 \\ 2 & 1\end{array}\right]$
(b) None of these
(c) $\frac{1}{3}\left[\begin{array}{ll}1 & 1 \\ 3 & 1\end{array}\right]$
(d) $\frac{1}{3}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
Answer: (c)
Explanation:
$
\begin{aligned}
&3 A=\left[\begin{array}{ll}
1 & 1 \\
3 & 1
\end{array}\right] \\
&A=\frac{1}{3}\left[\begin{array}{ll}
1 & 1 \\
3 & 1
\end{array}\right]
\end{aligned}
$

23) The transformation 'orthogonal projection on $X$-axis' is given by a matrix
(a) $\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]$
(b) $\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$ [0 $\left[\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right]$
Answer: (b)
Explanation: The orthogonal projection on $\mathrm{x}-$ axis is given by $\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]$

24) The system of equations, $x+y+z=1,3 x+6 y+z=8, \alpha x+2 y+3 z=1$ has a unique solution for
(a) All real $\alpha$
(b) $\alpha$ not equal to 0
(c) All integral $\alpha$
(d) All rational $\alpha$
Answer: (c)
Explanation:
The given system of equations has unique solution, if
$
\begin{aligned}
&\left|\begin{array}{lll}
1 & 1 & 1 \\
3 & 6 & 1 \\
\alpha & 2 & 3
\end{array}\right| \neq 0 \\
&\Rightarrow 1(18-2)-1(9-\alpha)+1(6-6 \alpha) \neq 0 \\
&\Rightarrow 13-5 \alpha \neq 0 \\
&\qquad \alpha \neq \frac{13}{5}
\end{aligned}
$

25) Two matrices A and B are multiplicative inverse of each other only if
(a) $\mathrm{AB}=\mathrm{BA}=0$
(b) $\quad \mathrm{AB}=0, \mathrm{BA}=\mathrm{I}$
(c) $\mathrm{AB}=\mathrm{BA}$
(d) $\quad \mathrm{AB}=\mathrm{BA}=\mathrm{I}$
Answer: (d)
Explanation:
If $A B=B A=I$, then $A$ and B are inverse of each other. i.e., A is inverse of $B$ and $B$ is inverse of $A$.