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 $\mathrm { A } = \left[ \begin{array} { l l l } 2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array} \right] , \mathrm { B } = \left[ \begin{array} { l l l } \mathrm { B } _ { 1 } & \mathrm {~B} _ { 2 } & \mathrm {~B} _ { 3 } \end{array} \right]$, where $\mathrm { B } _ { 1 } , \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 3 }$ are column matrices, and $\mathrm { AB } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, $\mathrm { AB } _ { 2 } = \left[ \begin{array} { l } 2 \\ 3 \\ 0 \end{array} \right] , \mathrm { AB } _ { 3 } = \left[ \begin{array} { l } 3 \\ 2 \\ 1 \end{array} \right]$ If $\alpha = | \mathrm { B } |$ and $\beta$ is the sum of all the diagonal elements of B, then $\alpha ^ { 3 } + \beta ^ { 3 }$ is equal to

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 $\_\_\_\_$

Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2 . If the roots of the equation $| A - x I | = 0$ be - 1 and 3 , then the sum of the diagonal elements of the matrix $A ^ { 2 }$ is $\_\_\_\_$ .

Let $A = \left[ \begin{array} { c c } 2 & - 1 \\ 1 & 1 \end{array} \right]$. If the sum of the diagonal elements of $A ^ { 13 }$ is $3 ^ { n }$, then $n$ is equal to $\_\_\_\_$

Let $A = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - 2 \\ 0 & 1 \end{array} \right]$ and $P = \left[ \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right] , \theta > 0$. If $\mathrm { B } = \mathrm { PAP } ^ { \mathrm { T } } , \mathrm { C } = \mathrm { P } ^ { \mathrm { T } } \mathrm { B } ^ { 10 } \mathrm { P }$ and the sum of the diagonal elements of $C$ is $\frac { \mathrm { m } } { \mathrm { n } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $\mathrm { m } + \mathrm { n }$ is :
(1) 127
(2) 258
(3) 65
(4) 2049

If the system of equations $$x + 2y - 3z = 2$$ $$2x + \lambda y + 5z = 5$$ $$14x + 3y + \mu z = 33$$ has infinitely many solutions, then $\lambda + \mu$ is equal to :
(1) 13
(2) 10
(3) 12
(4) 11

If the system of equations $$2x - y + z = 4$$ $$5x + \lambda y + 3z = 12$$ $$100x - 47y + \mu z = 212$$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to
(1) 57
(2) 59
(3) 55
(4) 56

If $\mathrm { A } , \mathrm { B }$, and $\left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right)$ are non-singular matrices of same order, then the inverse of $\mathrm { A } \left( \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) \right) ^ { - 1 } \mathrm {~B}$, is equal to
(1) $\mathrm { AB } ^ { - 1 } + \mathrm { A } ^ { - 1 } \mathrm {~B}$
(2) $\operatorname { adj } \left( \mathrm { B } ^ { - 1 } \right) + \operatorname { adj } \left( \mathrm { A } ^ { - 1 } \right)$
(3) $\frac { A B ^ { - 1 } } { | A | } + \frac { B A ^ { - 1 } } { | B | }$
(4) $\frac { 1 } { | A B | } ( \operatorname { adj } ( B ) + \operatorname { adj } ( A ) )$

Let $A = \left[ a _ { i j } \right]$ be a matrix of order $3 \times 3$, with $a _ { i j } = ( \sqrt { 2 } ) ^ { i + j }$. If the sum of all the elements in the third row of $A ^ { 2 }$ is $\alpha + \beta \sqrt { 2 } , \alpha , \beta \in \mathbf { Z }$, then $\alpha + \beta$ is equal to :
(1) 280
(2) 224
(3) 210
(4) 168

The system of equations $$x + y + z = 6$$ $$x + 2 y + 5 z = 9$$ $$x + 5 y + \lambda z = \mu$$ has no solution if
(1) $\lambda = 15 , \mu \neq 17$
(2) $\lambda \neq 17 , \mu \neq 18$
(3) $\lambda = 17 , \mu \neq 18$
(4) $\lambda = 17 , \mu = 18$

Let $\mathrm{A} = [\mathrm{a}_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}$. If $\mathrm{A}_{ij}$ is the cofactor of $\mathrm{a}_{ij}$, $\mathrm{C}_{ij} = \sum_{\mathrm{k}=1}^{2} \mathrm{a}_{i\mathrm{k}} \mathrm{A}_{j\mathrm{k}}$, $1 \leq i, j \leq 2$, and $\mathrm{C} = [\mathrm{C}_{ij}]$, then $8|\mathrm{C}|$ is equal to:
(1) 288
(2) 222
(3) 242
(4) 262

If the system of equations $$( \lambda - 1 ) x + ( \lambda - 4 ) y + \lambda z = 5$$ $$\lambda x + ( \lambda - 1 ) y + ( \lambda - 4 ) z = 7$$ $$( \lambda + 1 ) x + ( \lambda + 2 ) y - ( \lambda + 2 ) z = 9$$ has infinitely many solutions, then $\lambda ^ { 2 } + \lambda$ is equal to
(1) 6
(2) 10
(3) 20
(4) 12

Let M and m respectively be the maximum and the minimum values of $$f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{vmatrix}, \quad x \in \mathbf{R}.$$ Then $M^4 - m^4$ is equal to:
(1) 1280
(2) 1295
(3) 1215
(4) 1040

If the system of linear equations : $$x + y + 2z = 6$$ $$2x + 3y + \mathrm { a } z = \mathrm { a } + 1$$ $$- x - 3 y + \mathrm { b } z = 2 \mathrm {~b}$$ where $a , b \in \mathbf { R }$, has infinitely many solutions, then $7a + 3b$ is equal to :
(1) 16
(2) 12
(3) 22
(4) 9

Let $A = \left[ a _ { i j } \right]$ be $3 \times 3$ matrix such that $A \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] , A \left[ \begin{array} { l } 4 \\ 1 \\ 3 \end{array} \right] = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right]$ and $A \left[ \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right]$, then $a _ { 23 }$ equals :
(1) $- 1$
(2) 2
(3) 1
(4) 0

Let $A$ be a square matrix of order 3 such that $\operatorname { det } ( A ) = - 2$ and $\operatorname { det } ( 3 \operatorname { adj } ( - 6 \operatorname { adj } ( 3 A ) ) ) = 2 ^ { \mathrm { m } + \mathrm { n } } \cdot 3 ^ { \mathrm { mn } } , \mathrm { m } > \mathrm { n }$. Then $4 \mathrm {~m} + 2 \mathrm { n }$ is equal to $\_\_\_\_$

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm { S } = \{ - 3 , - 2 , - 1,1,2 \}$. Let
$$\begin{aligned} & \mathrm { S } _ { 1 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 2 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : \mathrm { A } = - \mathrm { A } ^ { \mathrm { T } } \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} , \\ & \mathrm { S } _ { 3 } = \left\{ \mathrm { A } = \left[ a _ { \mathrm { ij } } \right] \in \mathrm { M } : a _ { 11 } + a _ { 22 } + a _ { 33 } = 0 \text { and } a _ { \mathrm { ij } } \in \mathrm { S } , \forall \mathrm { i } , \mathrm { j } \right\} . \end{aligned}$$
If $n \left( \mathrm { S } _ { 1 } \cup \mathrm { S } _ { 2 } \cup \mathrm { S } _ { 3 } \right) = 125 \alpha$, then $\alpha$ equals

Let $A$ be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix},\quad A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix},$$ and $\det(\operatorname{adj}(2(A+I))) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is $\underline{\hspace{2cm}}$.

Let $\mathrm{S} = \{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2} + \mathrm{A}^{\mathrm{m}} = 3\mathrm{I} - \mathrm{A}^{-6}\}$, where $\mathrm{A} = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $\mathrm{n}(\mathrm{S})$ is equal to \_\_\_\_ .

Which of the following matrices is equal to $\left[ \begin{array} { c c } - 1 & 0 \\ 1 & - 1 \end{array} \right] ^ { 5 }$ ?
(1) $\left[ \begin{array} { c c } - 1 & 0 \\ - 5 & - 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ - 5 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } - 1 & 5 \\ 0 & - 1 \end{array} \right]$
(4) $\left[ \begin{array} { l l } 1 & 0 \\ 5 & 1 \end{array} \right]$
(5) $\left[ \begin{array} { c c } - 1 & 0 \\ 5 & - 1 \end{array} \right]$

Let matrix $A = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 } , B = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 6 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$, where $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$ is the inverse matrix of $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right]$. If $A + B = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, then $a + b + c + d =$ (10)(11).

Let $A = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$. If $A ^ { 4 } = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, what is the value of $a + b + c + d$?
(1) 158
(2) 162
(3) 166
(4) 170
(5) 174