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 $R = \left\{ \left( \begin{array} { c c c } a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{array} \right) : a , b , c , d \in \{ 0,3,5,7,11,13,17,19 \} \right\}$. Then the number of invertible matrices in $R$ is

Let $\alpha , \beta$ and $\gamma$ be real numbers. Consider the following system of linear equations
$x + 2 y + z = 7$
$x + \alpha z = 11$
$2 x - 3 y + \beta z = \gamma$
Match each entry in List-I to the correct entries in List-II.
List-I
(P) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma = 28$, then the system has
(Q) If $\beta = \frac { 1 } { 2 } ( 7 \alpha - 3 )$ and $\gamma \neq 28$, then the system has
(R) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma \neq 28$, then the system has
(S) If $\beta \neq \frac { 1 } { 2 } ( 7 \alpha - 3 )$ where $\alpha = 1$ and $\gamma = 28$, then the system has
List-II
(1) a unique solution
(2) no solution
(3) infinitely many solutions
(4) $x = 11 , y = - 2$ and $z = 0$ as a solution
(5) $x = - 15 , y = 4$ and $z = 0$ as a solution
The correct option is:
(A) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 4 )$
(B) $( P ) \rightarrow ( 3 ) \quad ( Q ) \rightarrow ( 2 ) \quad ( R ) \rightarrow ( 5 ) \quad ( S ) \rightarrow ( 4 )$
(C) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 4 ) \quad ( S ) \rightarrow ( 5 )$
(D) $( P ) \rightarrow ( 2 ) \quad ( Q ) \rightarrow ( 1 ) \quad ( R ) \rightarrow ( 1 ) \quad ( S ) \rightarrow ( 3 )$

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) $( \mathrm { P } ) \rightarrow ( 1 )$, $( \mathrm { Q } ) \rightarrow ( 5 )$, $( \mathrm { R } ) \rightarrow ( 3 )$, $( \mathrm { S } ) \rightarrow ( 4 )$

Let $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$. Let $Q = \left( \begin{array} { l l } x & y \\ z & 4 \end{array} \right)$ for some non-zero real numbers $x , y$, and $z$, for which there is a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers, such that $Q R = R P$.
Then which of the following statements is (are) TRUE?

(A)	The determinant of $Q - 2 I$ is zero
(B)	The determinant of $Q - 6 I$ is 12
(C)	The determinant of $Q - 3 I$ is 15
(D)	$y z = 2$

Let $A$ and $B$ be two symmetric matrices of order 3. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices. Statement-2: $AB$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

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) 0
(4) 3

Let $A$ and $B$ be real matrices of the form $\left[\begin{array}{ll}\alpha & 0 \\ 0 & \beta\end{array}\right]$ and $\left[\begin{array}{ll}0 & \gamma \\ \delta & 0\end{array}\right]$, respectively. Statement 1: $AB - BA$ is always an invertible matrix. Statement 2: $AB - BA$ is never an identity matrix.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is false, Statement 2 is true.
(3) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

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

If $A = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 2 & 1 & 0 \\ -3 & 2 & 1 \end{array} \right]$ and $B = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ -2 & 1 & 0 \\ 7 & -2 & 1 \end{array} \right]$ then $AB$ equals
(1) $I$
(2) $A$
(3) $B$
(4) $0$

If $A = \left( \begin{array} { c } \alpha - 1 \\ 0 \\ 0 \end{array} \right) , B = \left( \begin{array} { c } \alpha + 1 \\ 0 \\ 0 \end{array} \right)$ be two matrices, then $A B ^ { T }$ is a non-zero matrix for $| \alpha |$ not equal to
(1) 2
(2) 0
(3) 1
(4) 3

Statement 1: If the system of equations $x + k y + 3 z = 0, 3 x + k y - 2 z = 0, 2 x + 3 y - 4 z = 0$ has a nontrivial solution, then the value of $k$ is $\frac { 31 } { 2 }$. Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is true, Statement 2 is false.

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) $-1$

If the system of equations $$\begin{aligned} & x + y + z = 6 \\ & x + 2 y + 3 z = 10 \\ & x + 2 y + \lambda z = 0 \end{aligned}$$ has a unique solution, then $\lambda$ is not equal to
(1) 1
(2) 0
(3) 2
(4) 3

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

Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB - BA$ is
(1) skew symmetric
(2) $I$ or $- I$, where $I$ is an identity matrix
(3) symmetric
(4) neither symmetric nor skew symmetric

If $A$ is a $3 \times 3$ non-singular matrix such that $A A ^ { \prime } = A ^ { \prime } A$ and $B = A ^ { - 1 } A ^ { \prime }$, then $B B ^ { \prime }$ equals, where $X ^ { \prime }$ denotes the transpose of the matrix $X$.
(1) $B ^ { - 1 }$
(2) $\left( B ^ { - 1 } \right) ^ { \prime }$
(3) $I + B$
(4) $I$

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 $A = \begin{pmatrix} 5a & -b \\ 3 & 2 \end{pmatrix}$ and $A$ adj $A = A A^{T}$, then $5a + b$ is equal to: (1) $-1$ (2) $5$ (3) $4$ (4) $13$

The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for: (1) infinitely many values of $\lambda$ (2) exactly one value of $\lambda$ (3) exactly two values of $\lambda$ (4) exactly three values of $\lambda$

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A$ adj $A = A A^T$, then $5a + b$ is equal to:
(1) $-1$
(2) $5$
(3) $4$
(4) $13$

If $A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$, then adj$(3A^2 + 12A)$ is equal to: (1) $\begin{pmatrix} 72 & -84 \\ -63 & 51 \end{pmatrix}$ (2) $\begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix}$ (3) $\begin{pmatrix} 51 & 84 \\ 63 & 72 \end{pmatrix}$ (4) $\begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix}$

The system of linear equations \begin{align*} x + \lambda y - z &= 0 \lambda x - y - z &= 0 x + y - \lambda z &= 0 \end{align*} has a non-trivial solution for:
(1) infinitely many values of $\lambda$
(2) exactly one value of $\lambda$
(3) exactly two values of $\lambda$
(4) exactly three values of $\lambda$