If $A = \left[ \begin{array} { c c c } 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & - 1 & 3 \end{array} \right] , B = \mathrm{adj}\, A$ and $C = 3 A$, then $\frac { | \mathrm{adj}\, B | } { | C | }$ is equal to
(1) 8
(2) 16
(3) 72
(4) 2

If $x , y , z$ are in arithmetic progression with common difference $d , x \neq 3 d$, and the determinant of the matrix $\left[ \begin{array} { c c c } 3 & 4 \sqrt { 2 } & x \\ 4 & 5 \sqrt { 2 } & y \\ 5 & k & z \end{array} \right]$ is zero, then the value of $k ^ { 2 }$ is
(1) 72
(2) 12
(3) 36
(4) 6

Let $A = \left\{ a _ { i j } \right\}$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { l l } ( - 1 ) ^ { j - i } & \text { if } i < j \\ 2 & \text { if } i = j \\ ( - 1 ) ^ { i + j } & \text { if } i > j \end{array} \right.$ then $\det \left( 3 \operatorname{Adj} \left( 2 A ^ { - 1 } \right) \right)$ is equal to $\underline{\hspace{1cm}}$.

If $\mathrm { a } _ { \mathrm { r } } = \cos \frac { 2 \mathrm { r } \pi } { 9 } + i \sin \frac { 2 \mathrm { r } \pi } { 9 } , \mathrm { r } = 1,2,3 , \ldots , i = \sqrt { - 1 }$, then the determinant $\left| \begin{array} { l l l } a _ { 1 } & a _ { 2 } & a _ { 3 } \\ a _ { 4 } & a _ { 5 } & a _ { 6 } \\ a _ { 7 } & a _ { 8 } & a _ { 9 } \end{array} \right|$ is equal to $:$
(1) $\mathrm { a } _ { 9 }$
(2) $a _ { 1 } a _ { 9 } - a _ { 3 } a _ { 7 }$
(3) $a _ { 5 }$
(4) $a _ { 2 } a _ { 6 } - a _ { 4 } a _ { 8 }$

Let $A = \left[ \begin{array} { c c c } { [ x + 1 ] } & { [ x + 2 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 3 ] } & { [ x + 3 ] } \\ { [ x ] } & { [ x + 2 ] } & { [ x + 4 ] } \end{array} \right]$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. If $\operatorname { det } ( A ) = 192$, then the set of values of $x$ is in the interval: (1) $[ 62,63 )$ (2) $[ 65,66 )$ (3) $[ 60,61 )$ (4) $[ 68,69 )$

Let $A$ be a $3 \times 3$ invertible matrix. If $| \operatorname { adj } ( 24 A ) | = | \operatorname { adj } ( 3 \operatorname { adj } ( 2A ) ) |$, then $| A | ^ { 2 }$ is equal to
(1) $2 ^ { 6 }$
(2) $2 ^ { 12 }$
(3) 512
(4) $6 ^ { 6 }$

Let $A$ and $B$ be two $3 \times 3$ matrices such that $A B = I$ and $| A | = \frac { 1 } { 8 }$ then $| \operatorname{adj} ( B \operatorname{adj} ( 2 A ) ) |$ is equal to
(1) 128
(2) 32
(3) 64
(4) 102

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 $A$ be a $3 \times 3$ matrix and $\det(A) = 2$. If $n = \det(\text{adj}(\text{adj}(\cdots(\text{adj}(A))\cdots)))$ where adj is applied 6 times, then the remainder when $n$ is divided by 9 is $\_\_\_\_$.

Let $\mathrm { A } = \left[ \begin{array} { c c c } 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{array} \right]$ and $\mathrm { P } = \left[ \begin{array} { l l l } 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{array} \right]$. The sum of the prime factors of $\left| \mathrm { P } ^ { - 1 } \mathrm { AP } - 2 \mathrm { I } \right|$ is equal to
(1) 26
(2) 27
(3) 66
(4) 23

For a $3 \times 3$ matrix $M$, let trace ( $M$ ) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $| A | = \frac { 1 } { 2 }$ and trace $( A ) = 3$. If $B = \operatorname { adj } ( \operatorname { adj } ( 2 A ) )$, then the value of $| B | +$ trace $( \mathrm { B } )$ equals :
(1) 56
(2) 132
(3) 174
(4) 280