Let $A$ and $B$ be two $3 \times 3$ non-zero real matrices such that $A B$ is a zero matrix. Then
(1) The system of linear equations $A X = 0$ has a unique solution
(2) The system of linear equations $A X = 0$ has infinitely many solutions
(3) $B$ is an invertible matrix
(4) $\operatorname { adj } ( A )$ is an invertible matrix

Let $A = \left( \begin{array} { c c } 4 & - 2 \\ \alpha & \beta \end{array} \right)$. If $A ^ { 2 } + \gamma A + 18 I = O$, then $\operatorname { det } ( A )$ is equal to
(1) - 18
(2) 18
(3) - 50
(4) 50

Let $A = \left[ a _ { i j } \right]$ be a square matrix of order 3 such that $a _ { i j } = 2 ^ { j - i }$, for all $i , j = 1,2,3$. Then, the matrix $A ^ { 2 } + A ^ { 3 } + \ldots + A ^ { 10 }$ is equal to
(1) $\left( \frac { 3 ^ { 10 } - 1 } { 2 } \right) A$
(2) $\left( \frac { 3 ^ { 10 } + 1 } { 2 } \right) A$
(3) $\left( \frac { 3 ^ { 10 } + 3 } { 2 } \right) A$
(4) $\left( \frac { 3 ^ { 10 } - 3 } { 2 } \right) A$

Let $A = \left( \begin{array} { c c } 1 & 2 \\ - 2 & - 5 \end{array} \right)$. Let $\alpha , \beta \in \mathbb { R }$ be such that $\alpha A ^ { 2 } + \beta A = 2 I$. Then $\alpha + \beta$ is equal to
(1) - 10
(2) - 6
(3) 6
(4) 10

The number of real values of $\lambda$, such that the system of linear equations $2x - 3y + 5z = 9$ $x + 3y - z = -18$ $3x - y + (\lambda^2 - |\lambda|)z = 16$ has no solutions, is
(1) 0
(2) 1
(3) 2
(4) 4

Let $S$ be the set containing all $3 \times 3$ matrices with entries from $\{ - 1,0,1 \}$. The total number of matrices $A \in S$ such that the sum of all the diagonal elements of $A ^ { T } A$ is 6 is $\_\_\_\_$ .

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A ^ { 2 } + B = A ^ { 2 } B$, then
(1) $A B = I$
(2) $A ^ { 2 } B = I$
(3) $A ^ { 2 } = I$ or $B = I$
(4) $A ^ { 2 } B = B A ^ { 2 }$

Let $A , B , C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $( S 1 ) A ^ { 13 } B ^ { 26 } - B ^ { 26 } A ^ { 13 }$ is symmetric (S2) $A ^ { 26 } C ^ { 13 } - C ^ { 13 } A ^ { 26 }$ is symmetric Then,
(1) Only $S 2$ is true
(2) Only S1 is true
(3) Both $S 1$ and $S 2$ are false
(4) Both $S 1$ and $S 2$ are true

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^ { 2 } = 3 A + \alpha I$. If $A ^ { 4 } = 21 A + \beta I$, then
(1) $\alpha = 1$
(2) $\alpha = 4$
(3) $\beta = 8$
(4) $\beta = - 8$

Let $A = [a_{ij}]$, $a_{ij} \in Z \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $\_\_\_\_$.

Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$

Let $A = \left[ \begin{array} { c c } 1 & \frac { 1 } { 51 } \\ 0 & 1 \end{array} \right]$. If $B = \left[ \begin{array} { c c } 1 & 2 \\ - 1 & - 1 \end{array} \right] A \left[ \begin{array} { c c } - 1 & - 2 \\ 1 & 1 \end{array} \right]$, then the sum of all the elements of the matrix $\sum _ { n = 1 } ^ { 50 } B ^ { n }$ is equal to
(1) 75
(2) 125
(3) 50
(4) 100

Let $A$ be a $n \times n$ matrix such that $|A| = 2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2A^{-1}\right)\right)$ is $2^{84}$, then $n$ is equal to $\_\_\_\_$.

Let $P$ be a square matrix such that $P^{2} = I - P$. For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $P^{\alpha} + P^{\beta} = \gamma I - 29P$ and $P^{\alpha} - P^{\beta} = \delta I - 13P$, then $\alpha + \beta + \gamma - \delta$ is equal to
(1) 18
(2) 40
(3) 22
(4) 24

The set of all values of $t \in \mathbb { R }$, for which the matrix $\left[ \begin{array} { c c c } \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) & \mathrm { e } ^ { - t } ( - 2 \sin t - \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } ( 2 \sin t + \cos t ) & \mathrm { e } ^ { - t } ( \sin t - 2 \cos t ) \\ \mathrm { e } ^ { t } & \mathrm { e } ^ { - t } \cos t & \mathrm { e } ^ { - t } \sin t \end{array} \right]$ is invertible, is (1) $\left\{ ( 2 k + 1 ) \frac { \pi } { 2 } , k \in \mathbb { Z } \right\}$ (2) $\left\{ k \pi + \frac { \pi } { 4 } , k \in \mathbb { Z } \right\}$ (3) $\{ k \pi , k \in \mathbb { Z } \}$ (4) $\mathbb { R }$

If a point $P(\alpha, \beta, \gamma)$ satisfying $\begin{pmatrix} \alpha & \beta & \gamma \end{pmatrix} \begin{pmatrix} 2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ lies on the plane $2x + 4y + 3z = 5$, then $6\alpha + 9\beta + 7\gamma$ is equal to $\_\_\_\_$.

Let A be a symmetric matrix such that $| A | = 2$ and $\left[ \begin{array} { l l } 2 & 1 \\ 3 & \frac { 3 } { 2 } \end{array} \right] A = \left[ \begin{array} { l l } 1 & 2 \\ \alpha & \beta \end{array} \right]$. If the sum of the diagonal elements of A is $s$, then $\frac { \beta s } { \alpha ^ { 2 } }$ is equal to $\_\_\_\_$.

Let $A = \left[ \begin{array} { c c c } 2 & 1 & 0 \\ 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array} \right]$. If $| \mathrm{adj} ( \mathrm{adj} ( \mathrm{adj}\, 2 A ) ) | = ( 16 ) ^ { n }$, then $n$ is equal to
(1) 8
(2) 10
(3) 9
(4) 12

If $\alpha \neq \mathrm { a } , \beta \neq \mathrm { b } , \gamma \neq \mathrm { c }$ and $\left| \begin{array} { c c c } \alpha & \mathrm { b } & \mathrm { c } \\ \mathrm { a } & \beta & \mathrm { c } \\ \mathrm { a } & \mathrm { b } & \gamma \end{array} \right| = 0$, then $\frac { \mathrm { a } } { \alpha - \mathrm { a } } + \frac { \mathrm { b } } { \beta - \mathrm { b } } + \frac { \gamma } { \gamma - \mathrm { c } }$ is equal to: (1) 3 (2) 0 (3) 1 (4) 2

Let A be a square matrix such that $\mathrm { AA } ^ { \mathrm { T } } = \mathrm { I }$. Then $\frac { 1 } { 2 } \mathrm { ~A} \left[ \left( \mathrm { ~A} + \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } + \left( \mathrm { A } - \mathrm { A } ^ { \mathrm { T } } \right) ^ { 2 } \right]$ is equal to
(1) $A ^ { 2 } + I$
(2) $A ^ { 3 } + I$
(3) $A ^ { 2 } + A ^ { T }$
(4) $\mathrm { A } ^ { 3 } + \mathrm { A } ^ { \mathrm { T } }$

If $A$ is a square matrix of order 3 such that $\operatorname { det } ( A ) = 3$ and $\operatorname { det } \left( \operatorname { adj } \left( - 4 \operatorname { adj } \left( - 3 \operatorname { adj } \left( 3 \operatorname { adj } \left( ( 2 \mathrm {~A} ) ^ { - 1 } \right) \right) \right) \right) \right) = 2 ^ { \mathrm { m } } 3 ^ { \mathrm { n } }$, then $\mathrm { m } + 2 \mathrm { n }$ is equal to:
(1) 2
(2) 3
(3) 6
(4) 4

Let $B = \left[ \begin{array} { l l } 1 & 3 \\ 1 & 5 \end{array} \right]$ and $A$ be a $2 \times 2$ matrix such that $A B ^ { - 1 } = A ^ { - 1 }$. If $B C B ^ { - 1 } = A$ and $C ^ { 4 } + \alpha C ^ { 2 } + \beta I = O$, then $2 \beta - \alpha$ is equal to
(1) 16
(2) 2
(3) 8
(4) 10

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]$ and $B = I + \operatorname { adj } ( A ) + ( \operatorname { adj } A ) ^ { 2 } + \ldots + ( \operatorname { adj } A ) ^ { 10 }$. Then, the sum of all the elements of the matrix $B$ is:
(1) - 124
(2) 22
(3) - 88
(4) - 110

Consider the matrix $f ( x ) = \left[ \begin{array} { c c c } \cos x & - \sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{array} \right]$. Given below are two statements: Statement I: $f ( - x )$ is the inverse of the matrix $f ( x )$. Statement II: $f ( x ) f ( y ) = f ( x + y )$. In the light of the above statements, choose the correct answer from the options given below
(1) Statement I is false but Statement II is true
(2) Both Statement I and Statement II are false
(3) Statement I is true but Statement II is false
(4) Both Statement I and Statement II are true

Let $A$ be a $2 \times 2$ symmetric matrix such that $A \left[ \begin{array} { l } 1 \\ 1 \end{array} \right] = \left[ \begin{array} { l } 3 \\ 7 \end{array} \right]$ and the determinant of $A$ be 1 . If $A ^ { - 1 } = \alpha A + \beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha + \beta$ equals $\_\_\_\_$