Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^ { - 1 } = \operatorname { adj } ( \operatorname { adj } M )$, then which of the following statements is/are ALWAYS TRUE?
(A) $M = I$
(B) $\operatorname { det } M = 1$
(C) $M ^ { 2 } = I$
(D) $( \operatorname { adj } M ) ^ { 2 } = I$

Let $M = \begin{pmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{pmatrix}$ and $\text{adj}(M) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
where $a$ and $b$ are real numbers. Which of the following statements is(are) TRUE?
(A) $(a+b)^2 = 9$
(B) $\det(\text{adj}(M^2)) = 81$
(C) $\text{adj}(\text{adj}(M)) = \begin{pmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{pmatrix}$
(D) $\det(\text{adj}(2M)) = 2^8$

Let $M = \left( a _ { i j } \right) , i , j \in \{ 1,2,3 \}$, be the $3 \times 3$ matrix such that $a _ { i j } = 1$ if $j + 1$ is divisible by $i$, otherwise $a _ { i j } = 0$. Then which of the following statements is(are) true?
(A) $M$ is invertible
(B) There exists a nonzero column matrix $\left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right)$ such that $M \left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right) = \left( \begin{array} { l } - a _ { 1 } \\ - a _ { 2 } \\ - a _ { 3 } \end{array} \right)$
(C) The set $\left\{ X \in \mathbb { R } ^ { 3 } : M X = \mathbf { 0 } \right\} \neq \{ \mathbf { 0 } \}$, where $\mathbf { 0 } = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)$
(D) The matrix $( M - 2 I )$ is invertible, where $I$ is the $3 \times 3$ identity matrix

Let $S = \left\{ A = \left( \begin{array} { l l l } 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{array} \right) : a , b , c , d , e \in \{ 0,1 \} \right.$ and $\left. | A | \in \{ - 1,1 \} \right\}$, where $| A |$ denotes the determinant of $A$. Then the number of elements in $S$ is $\_\_\_\_$ .

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x ^ { 2 } + x - 1 = 0$. Consider the set $T = \{ 1 , \alpha , \beta \}$. For a $3 \times 3$ matrix $M = \left( a _ { i j } \right) _ { 3 \times 3 }$, define $R _ { i } = a _ { i 1 } + a _ { i 2 } + a _ { i 3 }$ and $C _ { j } = a _ { 1 j } + a _ { 2 j } + a _ { 3 j }$ for $i = 1,2,3$ and $j = 1,2,3$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) The number of matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $R _ { i } = C _ { j } = 0$ for all $i , j$, is
(Q) The number of symmetric matrices $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ with all entries in $T$ such that $C _ { j } = 0$ for all $j$, is
(R) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a skew symmetric matrix such that $a _ { i j } \in T$ for $i > j$. Then the number of elements in the set $\left\{ \left( \begin{array} { l } x \\ y \\ z \end{array} \right) : x , y , z \in \mathbb { R } , M \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } a _ { 12 } \\ 0 \\ - a _ { 23 } \end{array} \right) \right\}$ is
(S) Let $M = \left( a _ { i j } \right) _ { 3 \times 3 }$ be a matrix with all entries in $T$ such that $R _ { i } = 0$ for all $i$. Then the absolute value of the determinant of $M$ is
List-II
(1) 1
(2) 12
(3) infinite
(4) 6
(5) 0
The correct option is
(A) $( \mathrm { P } ) \rightarrow ( 4 )$, $( \mathrm { Q } ) \rightarrow ( 2 )$, $( \mathrm { R } ) \rightarrow ( 5 )$, $( \mathrm { S } ) \rightarrow ( 1 )$
(B) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 1 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(C) $( \mathrm { P } ) \rightarrow ( 2 )$, $( \mathrm { Q } ) \rightarrow ( 4 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 5 )$
(D) $( \mathrm { P } ) \rightarrow ( 1 )$, $( \mathrm { Q } ) \rightarrow ( 5 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 4 )$

If $D = \left| \begin{array} { c c c } 1 & 1 & 1 \\ 1 & 1 + x & 1 \\ 1 & 1 & 1 + y \end{array} \right|$ for $x \neq 0 , y \neq 0$ then $D$ is
(1) divisible by neither $x$ nor $y$
(2) divisible by both $x$ and $y$
(3) divisible by $x$ but not $y$
(4) divisible by $y$ but not $x$

$$\left|\begin{array}{ccc} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & b+c & -2c \end{array}\right| = \alpha(a+b)(b+c)(c+a) \neq 0$$
then $\alpha$ is equal to
(1) $a+b+c$
(2) $abc$
(3) 4
(4) 1

Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^{3} = Q^{3}$ and $P^{2}Q = Q^{2}P$, then the determinant of $(P^{2}+Q^{2})$ is equal to
(1) $-2$
(2) 1
(3) 0
(4) $-1$

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}$. If $u_{1}$ and $u_{2}$ are column matrices such that $Au_{1} = \begin{pmatrix}1\\0\\0\end{pmatrix}$ and $Au_{2} = \begin{pmatrix}0\\1\\0\end{pmatrix}$, then $u_{1}+u_{2}$ is equal to
(1) $\begin{pmatrix}-1\\1\\0\end{pmatrix}$
(2) $\begin{pmatrix}-1\\1\\-1\end{pmatrix}$
(3) $\begin{pmatrix}-1\\-1\\0\end{pmatrix}$
(4) $\begin{pmatrix}1\\-1\\-1\end{pmatrix}$

The number of values of $k$ for which the linear equations $4x+ky+2z=0$, $kx+4y+z=0$, $2x+2y+z=0$ possess a non-zero solution is
(1) 2
(2) 1
(3) zero
(4) 3

If $\alpha , \beta \neq 0 , f ( n ) = \alpha ^ { n } + \beta ^ { n }$ and $$\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K ( 1 - \alpha ) ^ { 2 } ( 1 - \beta ) ^ { 2 } ( \alpha - \beta ) ^ { 2 }$$, then $K$ is equal to
(1) 1
(2) - 1
(3) $\alpha \beta$
(4) $\frac { 1 } { \alpha \beta }$

If $B$ is a $3 \times 3$ matrix such that $B ^ { 2 } = 0$, then det. $\left[ ( I + B ) ^ { 50 } - 50 B \right]$ is equal to:
(1) 1
(2) 2
(3) 3
(4) 50

The mean of 5 observations is 5 and their variance is 12.4 . If three of the observations are $1,2 \& 6$; then the value of the remaining two is :
(1) 1,11
(2) 5,5
(3) 5,11
(4) None of these

Let $A$, be a $3 \times 3$ matrix, such that $A ^ { 2 } - 5 A + 7 I = O$. Statement - I : $A ^ { - 1 } = \frac { 1 } { 7 } ( 5 I - A )$. Statement - II : The polynomial $A ^ { 3 } - 2 A ^ { 2 } - 3 A + I$, can be reduced to $5 ( A - 4 I )$. Then :
(1) Both the statements are true
(2) Both the statements are false
(3) Statement - I is true, but Statement - II is false
(4) Statement - I is false, but Statement - II is true

If $A = \left[ \begin{array} { c c } - 4 & - 1 \\ 3 & 1 \end{array} \right]$, then the determinant of the matrix $\left( A ^ { 2016 } - 2 A ^ { 2015 } - A ^ { 2014 } \right)$ is :
(1) $- 175$
(2) 2014
(3) 2016
(4) $- 25$

If $\alpha$, $\beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and $$\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,$$ then $K$ is equal to: (1) $\alpha\beta$ (2) $\frac{1}{\alpha\beta}$ (3) 1 (4) $-1$

The set of all values of $\lambda$ for which the system of linear equations $$x - 2 y - 2 z = \lambda x$$ $$x + 2 y + z = \lambda y$$ $$- x - y = \lambda z$$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

Let $A$ be any $3 \times 3$ invertible matrix. Then which one of the following is not always true?
(1) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | ^ { 2 } \cdot ( \operatorname { adj } ( \mathrm {~A} ) ) ^ { - 1 }$
(2) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot ( \operatorname { adj } ( \mathrm { A } ) ) ^ { - 1 }$
(3) $\operatorname { adj } ( \operatorname { adj } ( \mathrm { A } ) ) = | A | \cdot A$
(4) $\operatorname { adj } ( \mathrm { A } ) = | A | \cdot A ^ { - 1 }$

The number of real values of $\lambda$ for which the system of linear equations, $2 x + 4 y - \lambda z = 0, 4 x + \lambda y + 2 z = 0$ and $\lambda x + 2 y + 2 z = 0$, has infinitely many solutions, is:
(1) 3
(2) 1
(3) 2
(4) 0

If $S = \left\{ x \in [ 0,2 \pi ] : \left| \begin{array} { c c c } 0 & \cos x & - \sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{array} \right| = 0 \right\}$, then $\sum _ { x \in S } \tan \left( \frac { \pi } { 3 } + x \right)$ is equal to:
(1) $4 + 2 \sqrt { 3 }$
(2) $- 4 - 2 \sqrt { 3 }$
(3) $- 2 + \sqrt { 3 }$
(4) $- 2 - \sqrt { 3 }$

$\left| \begin{array} { c c c } x - 4 & 2 x & 2 x \\ 2 x & x - 4 & 2 x \\ 2 x & 2 x & x - 4 \end{array} \right| = ( A + B x ) ( x - A ) ^ { 2 }$, then the ordered pair $( A , B )$ is equal to
(1) $( 4,5 )$
(2) $( - 4 , - 5 )$
(3) $( - 4,3 )$
(4) $( - 4,5 )$

If $A = \left[\begin{array}{ccc} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{array}\right]$, then $A$ is:
(1) Invertible only if $t = \pi$
(2) Not invertible for any $t \in R$
(3) Invertible only if $t = \frac{\pi}{2}$
(4) Invertible for all $t \in R$

If $A = \left[ \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right]$, then for all $\theta \in \left( \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } \right) , \operatorname { det } ( A )$ lies in the interval :
(1) $\left( 1 , \frac { 5 } { 2 } \right]$
(2) $\left[ \frac { 5 } { 2 } , 4 \right)$
(3) $\left( \frac { 3 } { 2 } , 3 \right]$
(4) $\left( 0 , \frac { 3 } { 2 } \right]$

The set of all values of $\lambda$ for which the system of linear equations $$\begin{aligned} & x - 2 y - 2 z = \lambda x \\ & x + 2 y + z = \lambda y \\ & - x - y = \lambda z \end{aligned}$$ has a non-trivial solution :
(1) is an empty set
(2) contains more than two elements
(3) is a singleton
(4) contains exactly two elements

If the system of equations $2 x + 3 y - z = 0 , x + k y - 2 z = 0$ and $2 x - y + z = 0$ has a non-trivial solution $( x , y , z )$, then $\frac { x } { y } + \frac { y } { z } + \frac { z } { x } + k$ is equal to
(1) $- \frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) $- 4$
(4) $\frac { 3 } { 4 }$