Let $z = \frac{-1 + \sqrt{3}\,i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1,2,3\}$. Let $P = \left[\begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2 = -I$ is

Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?
[A] $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
[B] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[C] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[D] $\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$

How many $3 \times 3$ matrices $M$ with entries from $\{ 0,1,2 \}$ are there, for which the sum of the diagonal entries of $M ^ { T } M$ is 5 ?
[A] 126
[B] 198
[C] 162
[D] 135

Let $$P_1 = I = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_2 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], \quad P_3 = \left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right],$$ $$P_4 = \left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right], \quad P_5 = \left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right], \quad P_6 = \left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$$ and $X = \sum_{k=1}^{6} P_k \left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?
(A) If $X\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] = \alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha = 30$
(B) $X$ is a symmetric matrix
(C) The sum of diagonal entries of $X$ is 18
(D) $X - 30I$ is an invertible matrix

Let $$M = \left[ \begin{array} { c c } \sin ^ { 4 } \theta & - 1 - \sin ^ { 2 } \theta \\ 1 + \cos ^ { 2 } \theta & \cos ^ { 4 } \theta \end{array} \right] = \alpha I + \beta M ^ { - 1 }$$ where $\alpha = \alpha ( \theta )$ and $\beta = \beta ( \theta )$ are real numbers, and $I$ is the $2 \times 2$ identity matrix. If $\alpha ^ { * }$ is the minimum of the set $\{ \alpha ( \theta ) : \theta \in [ 0,2 \pi ) \}$ and $\beta ^ { * }$ is the minimum of the set $\{ \beta ( \theta ) : \theta \in [ 0,2 \pi ) \}$, then the value of $\alpha ^ { * } + \beta ^ { * }$ is
(A) $- \frac { 37 } { 16 }$
(B) $- \frac { 31 } { 16 }$
(C) $- \frac { 29 } { 16 }$
(D) $- \frac { 17 } { 16 }$

Let $x \in \mathbb{R}$ and let $$P = \left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3\end{array}\right], \quad Q = \left[\begin{array}{ccc}2 & x & x \\ 0 & 4 & 0 \\ x & x & 6\end{array}\right] \text{ and } R = PQP^{-1}$$
Then which of the following options is/are correct?
(A) There exists a real number $x$ such that $PQ = QP$
(B) $\det R = \det\left[\begin{array}{lll}2 & x & x \\ 0 & 4 & 0 \\ x & x & 5\end{array}\right] + 8$, for all $x \in \mathbb{R}$
(C) For $x = 0$, if $R\left[\begin{array}{l}1 \\ a \\ b\end{array}\right] = 6\left[\begin{array}{l}1 \\ a \\ b\end{array}\right]$, then $a + b = 5$
(D) For $x = 1$, there exists a unit vector $\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$ for which $R\left[\begin{array}{l}\alpha \\ \beta \\ \gamma\end{array}\right] = \left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

Let $$M = \left[ \begin{array} { l l l } 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{array} \right] \quad \text { and } \quad \operatorname { adj } M = \left[ \begin{array} { r r r } - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{array} \right]$$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?
(A) $a + b = 3$
(B) $\quad ( \operatorname { adj } M ) ^ { - 1 } + \operatorname { adj } M ^ { - 1 } = - M$
(C) $\operatorname { det } \left( \operatorname { adj } M ^ { 2 } \right) = 81$
(D) If $M \left[ \begin{array} { l } \alpha \\ \beta \\ \gamma \end{array} \right] = \left[ \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right]$, then $\alpha - \beta + \gamma = 3$

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $-18$, then the value of the determinant of $A$ is $\_\_\_\_$

Let $\beta$ be a real number. Consider the matrix
$$A = \left( \begin{array} { c c c } \beta & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{array} \right)$$
If $A ^ { 7 } - ( \beta - 1 ) A ^ { 6 } - \beta A ^ { 5 }$ is a singular matrix, then the value of $9 \beta$ is $\_\_\_\_$ .

Let $| M |$ denote the determinant of a square matrix $M$. Let $g : \left[ 0 , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be the function defined by
$$g ( \theta ) = \sqrt { f ( \theta ) - 1 } + \sqrt { f \left( \frac { \pi } { 2 } - \theta \right) - 1 }$$
where $$f ( \theta ) = \frac { 1 } { 2 } \left| \begin{array} { c c c } 1 & \sin \theta & 1 \\ - \sin \theta & 1 & \sin \theta \\ - 1 & - \sin \theta & 1 \end{array} \right| + \left| \begin{array} { c c c } \sin \pi & \cos \left( \theta + \frac { \pi } { 4 } \right) & \tan \left( \theta - \frac { \pi } { 4 } \right) \\ \sin \left( \theta - \frac { \pi } { 4 } \right) & - \cos \frac { \pi } { 2 } & \log _ { e } \left( \frac { 4 } { \pi } \right) \\ \cot \left( \theta + \frac { \pi } { 4 } \right) & \log _ { e } \left( \frac { \pi } { 4 } \right) & \tan \pi \end{array} \right|$$
Let $p ( x )$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g ( \theta )$, and $p ( 2 ) = 2 - \sqrt { 2 }$. Then, which of the following is/are TRUE ?
(A) $p \left( \frac { 3 + \sqrt { 2 } } { 4 } \right) < 0$
(B) $p \left( \frac { 1 + 3 \sqrt { 2 } } { 4 } \right) > 0$
(C) $p \left( \frac { 5 \sqrt { 2 } - 1 } { 4 } \right) > 0$
(D) $\quad p \left( \frac { 5 - \sqrt { 2 } } { 4 } \right) < 0$

If $M = \left( \begin{array} { r r } \frac { 5 } { 2 } & \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$, then which of the following matrices is equal to $M ^ { 2022 }$ ?
(A) $\quad \left( \begin{array} { r r } 3034 & 3033 \\ - 3033 & - 3032 \end{array} \right)$
(B) $\quad \left( \begin{array} { l l } 3034 & - 3033 \\ 3033 & - 3032 \end{array} \right)$
(C) $\quad \left( \begin{array} { r r } 3033 & 3032 \\ - 3032 & - 3031 \end{array} \right)$
(D) $\quad \left( \begin{array} { r r } 3032 & 3031 \\ - 3031 & - 3030 \end{array} \right)$

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

Consider the matrix
$$P = \left( \begin{array} { l l l } 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array} \right)$$
Let the transpose of a matrix $X$ be denoted by $X ^ { T }$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that
$$Q ^ { - 1 } = Q ^ { T } \text { and } P Q = Q P$$
is

(A)

32

(B)

8

(C)

16

(D)

24

Let $A = \left[ \begin{array} { c c c } 5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\ 0 & 0 & 5 \end{array} \right]$. If $\left| A ^ { 2 } \right| = 25$, then $| \alpha |$ equals
(1) $5 ^ { 2 }$
(2) 1
(3) $1 / 5$
(4) 5

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.

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.

If $P = \left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to
(1) 5
(2) 0
(3) 4
(4) 11

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 $\Delta _ { r } = \left| \begin{array} { c c c } r & 2 r - 1 & 3 r - 2 \\ \frac { n } { 2 } & n - 1 & a \\ \frac { 1 } { 2 } n ( n - 1 ) & ( n - 1 ) ^ { 2 } & \frac { 1 } { 2 } ( n - 1 ) ( 3 n + 4 ) \end{array} \right|$, then the value of $\sum _ { r = 1 } ^ { n - 1 } \Delta _ { r }$
(1) Is independent of both $a$ and $n$
(2) Depends only on $a$
(3) Depends only on $n$
(4) Depends both on $a$ and $n$

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) $I$

The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is:
(1) 0
(2) 2
(3) 1
(4) 3