Matrices — Class 12 Mathematics PYQs

If $A=\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$, then $A^{-1}$ is

(A) $\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

(B) $\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right]$

(C) $\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]$

(D) $\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=j-2i$. Then which of the following is true ?

(A) $a_{12}>0$
(B) all $a_{ij}<0$
(C) $a_{13}+a_{31}=-6$
(D) $a_{23}>a_{32}$

Let $A$ be a matrix of order $m\times n$ and $B$ is a matrix such that $A^{T}B$ and $B{A}^T$ are defined. Then, the order of $B$ is :

(A) $m\times m$
(B) $n\times n$
(C) $m\times n$
(D) $n\times m$

Let $A=\left[\begin{array}{ccc}1& -2& -1\\ 0& 4& -1\\ -3& 2& 1\end{array}\right]$, $B=\left[\begin{array}{c}-2\\ -5\\ -7\end{array}\right]$, $C=\left[\begin{array}{ccc}9& 8& 7\end{array}\right]$, which of the following is defined ?

(A) Only $AB$
(B) Only $AC$
(C) Only $BA$
(D) All $AB$, $AC$ and $BA$

If $A=\left[\begin{array}{ccc}7& 0& x\\ 0& 7& 0\\ 0& 0& y\end{array}\right]$ is a scalar matrix, then $y^x$ is equal to

(A) $0$
(B) $1$
(C) $7$
(D) $\pm 7$

If $A$ and $B$ are invertible matrices, then which of the following is not correct ?

(A) $(A+B{)}^{-1}=B^{-1}+A^{-1}$
(B) $(AB{)}^{-1}=B^{-1}{A}^{-1}$
(C) $\text{adj}(A)=|A|\cdot A^{-1}$
(D) $|A{|}^{-1}=|A^{-1}|$

A system of linear equations is represented as $AX=B$, where $A$ is coefficient matrix, $X$ is variable matrix and $B$ is the constant matrix. Then the system of equations is

(A) Consistent, if $|A|\ne 0$, solution is given by $X=B{A}^{-1}$.
(B) Inconsistent if $|A|=0$ and $(\text{adj }A)B=0$
(C) Inconsistent if $|A|\ne 0$
(D) May or may not be consistent if $|A|=0$ and $(\text{adj }A)B=0$

If $A=\left[\begin{array}{ccc}1& 12& 4y\\ 6x& 5& 2x\\ 8x& 4& 6\end{array}\right]$ is a symmetric matrix, then $(2x+y)$ is

(A) $-8$
(B) $0$
(C) $6$
(D) $8$

Which of the following can be both a symmetric and skew-symmetric matrix ?

(A) Unit Matrix
(B) Diagonal Matrix
(C) Null Matrix
(D) Row Matrix

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify $4AB+3(AB+BA)-4BA$, where $A$ and $B$ are both matrices of order $2\times 2$. It is known that $A\ne B\ne I$ and $A^{-1}\ne B$.

Their answers are given as :
Abhay : $6AB$
Bina : $7AB-BA$
Chhaya : $8AB$
Devesh : $7BA-AB$

Who answered it correctly ?

(A) Abhay
(B) Bina
(C) Chhaya
(D) Devesh