If $A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{pmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to:
(1) $(2, -1)$
(2) $(-2, 1)$
(3) $(2, 1)$
(4) $(-2, -1)$

$A = \left[ \begin{array} { c c c } 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array} \right]$ is a matrix satisfying the equation $A A ^ { T } = 9 I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $( a , b )$ is equal to
(1) $( - 2 , - 1 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( 2,1 )$

The set of all values of $\lambda$ for which the system of linear equations: $2 x _ { 1 } - 2 x _ { 2 } + x _ { 3 } = \lambda x _ { 1 }$ $2 x _ { 1 } - 3 x _ { 2 } + 2 x _ { 3 } = \lambda x _ { 2 }$ $- x _ { 1 } + 2 x _ { 2 } = \lambda x _ { 3 }$ has a non-trivial solution,
(1) Contains more than two elements.
(2) Is an empty set.
(3) Is a singleton.
(4) Contains two elements.

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 $\lambda$

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

If $A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$, then $\text{Adj}(3A^2 + 12A)$ 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 number of real values of $\lambda$ for which the system of linear equations $$2 x + 4 y - \lambda z = 0$$ $$4 x + \lambda y + 2 z = 0$$ $$\lambda x + 2 y + 2 z = 0$$ has infinitely many solutions, is:
(1) 0
(2) 1
(3) 2
(4) 3

Let $A$ be a matrix such that $A \cdot \left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then, $A ^ { 2 }$ equals :
(1) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$

Let $A$ be a matrix such that $A$. $\left[ \begin{array} { l l } 1 & 2 \\ 0 & 3 \end{array} \right]$ is a scalar matrix and $| 3 A | = 108$. Then $A ^ { 2 }$ equals
(1) $\left[ \begin{array} { c c } 4 & - 32 \\ 0 & 36 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 4 & 0 \\ - 32 & 36 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 36 & 0 \\ - 32 & 4 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 36 & - 32 \\ 0 & 4 \end{array} \right]$

If $\left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right] \ldots \left[ \begin{array} { c c } 1 & n - 1 \\ 0 & 1 \end{array} \right] = \left[ \begin{array} { c c } 1 & 78 \\ 0 & 1 \end{array} \right]$, then the inverse of $\left[ \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right]$ is:
(1) $\left[ \begin{array} { c c } 1 & - 12 \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ 12 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } 1 & 0 \\ 13 & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & - 13 \\ 0 & 1 \end{array} \right]$

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $\alpha \in R$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then, a value of $\alpha$ is:
(1) 0
(2) $\frac{\pi}{16}$
(3) $\frac{\pi}{64}$
(4) $\frac{\pi}{32}$

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } + x + 1 = 0$. Then for $y \neq 0$ in $R , \left| \begin{array} { c c c } y + 1 & \alpha & \beta \\ \alpha & y + \beta & 1 \\ \beta & 1 & y + \alpha \end{array} \right|$ is equal to
(1) $y ^ { 3 }$
(2) $y \left( y ^ { 2 } - 1 \right)$
(3) $y ^ { 3 } - 1$
(4) $y \left( y ^ { 2 } - 3 \right)$

Let the numbers $2 , b , c$ be in an A.P. and $A = \begin{pmatrix} 2 & b & c \\ 4 & b^2 & c^2 \end{pmatrix}$. If $\det ( A ) \in [ 2,16 ]$, then $c$ lies in the interval:
(1) $[2,3]$
(2) $[4,6]$
(3) $\left[3, 2 + 2 ^ { \frac { 3 } { 4 } }\right]$
(4) $\left[2 + 2 ^ { \frac { 3 } { 4 } } , 4\right]$

The total number of matrices $A = \left( \begin{array} { c c c } 0 & 2 y & 1 \\ 2 x & y & - 1 \\ 2 x & - y & 1 \end{array} \right) , ( x , y \in R , x \neq y )$ for which $A ^ { T } A = 3 I _ { 3 }$ is:
(1) 6
(2) 3
(3) 4
(4) 2

If $A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix}$ and $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, then $10A^{-1}$ is equal to.
(1) $A - 4I$
(2) $6I - A$
(3) $A - 6I$
(4) $4I - A$

The following system of linear equations $7 x + 6 y - 2 z = 0$ $3 x + 4 y + 2 z = 0$ $x - 2 y - 6 z = 0$, has
(1) infinitely many solutions, ( $x , y , z$ ) satisfying $y = 2z$
(2) no solution
(3) infinitely many solutions, $( x , y , z )$ satisfying $x = 2z$
(4) only the trivial solution

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements: $(P)$ If $A \neq I_{2}$, then $|A| = -1$ $(Q)$ If $|A| = 1$, then $\operatorname{tr}(A) = 2$ Where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then
(1) $(P)$ is false and $(Q)$ is true
(2) Both $(P)$ and $(Q)$ are false
(3) $(P)$ is true and $(Q)$ is false
(4) Both $(P)$ and $(Q)$ are true

Let $\theta=\frac{\pi}{5}$ and $A=\left[\begin{array}{cc}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{array}\right]$. If $B=A+A^{4}$, then $\det(B)$:
(1) is one
(2) lies in $(2,3)$
(3) is zero
(4) lies in $(1,2)$

Let $\mathrm { A } = \left[ a _ { i j } \right]$ and $\mathrm { B } = \left[ b _ { i j } \right]$ be two $3 \times 3$ real matrices such that $b _ { i j } = ( 3 ) ^ { ( i + j - 2 ) } a _ { i j }$, where $i , j = 1,2,3$. If the determinant of B is 81, then determinant of A is
(1) $\frac { 1 } { 3 }$
(2) 3
(3) $\frac { 1 } { 81 }$
(4) $\frac { 1 } { 9 }$

Let $a - 2 b + c = 1$. If $f ( x ) = \left| \begin{array} { l l l } x + a & x + 2 & x + 1 \\ x + b & x + 3 & x + 2 \\ x + c & x + 4 & x + 3 \end{array} \right|$, then:
(1) $f ( - 50 ) = 501$
(2) $f ( - 50 ) = - 1$
(3) $f ( 50 ) = - 501$
(4) $f ( 50 ) = 1$

If $\Delta = \left| \begin{array} { c c c } x - 2 & 2 x - 3 & 3 x - 4 \\ 2 x - 3 & 3 x - 4 & 4 x - 5 \\ 3 x - 5 & 5 x - 8 & 10 x - 17 \end{array} \right| = A x ^ { 3 } + B x ^ { 2 } + C x + D$, then $B + C$ is equal to:
(1) $- 1$
(2) $1$
(3) $- 3$
(4) $9$