Let $A={\left[a_{ij}\right]}_{2\times 2},$ where $a_{ij=0}$ for all $i,j$ and $A^2=I.$ Let a be the sum of all diagonal elements of A and b = |A|. Then $3a^2+4b^2$ is equal to

(a) 3                              (b) 7                           (c) 4                            (d) 14

Let P be a square matrix such that $P^2=I-P.$ For $\alpha ,\beta ,\gamma ,\delta \in \mathrm{\mathbb{N}},$ if $P^{\alpha }+P^{\beta }=\gamma I-29P$

and $P^{\alpha }-P^{\beta }=\delta I-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

(a) 40                          (b) 22                        (c) 18                         (d) 24

Let $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right],A=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ and $Q=PA{P}^T.$ If $P^T$ $Q^{2007}$ $P=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$ then $2a+b-3c-4d$ equal to

(a) 2006                          (b) 2004                              (c) 2005                               (d) 2007

Let $A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right].$ If $B=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]$ $A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix $\sum _{n=1}^{50}{B}^n$ is equal to

(a) 50                         (b) 100                              (c) 75                         (d) 125

The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is

(a) ${10}^6$                        (b) ${10}^9$                      (c) $6^{10}$                          (d) $9^{10}$

If $A=\frac{1}{2}\left[\begin{array}{cc}1& \sqrt{3}\\ -\sqrt{3}& 1\end{array}\right],$ then

(a) $A^{30}-A^{25}=2I$                     (b) $A^{30}=A^{25}$                   (c) $A^{30}+A^{25}-A=I$                      (d) $A^{30}+A^{25}+A=I$

If A and B are two non-zero $n\times n$ matrices such that $A^2+B=A^{2}B,$ then

(a) $A^2=I or B = I$                    (b) $A^{2}B=B{A}^2$                       (c) $AB=I$                     (D) $A^{2}B=I$

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}}& \frac{3}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}& \frac{1}{\sqrt{10}}\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -i\\ 0& 1\end{array}\right]$, where $i=\sqrt{-1}.$ If $M=A^{T}B$ $A,$ then the inverse of the matrix $A{M}^{2023}$ $A^T$ is

(a) $\left[\begin{array}{cc}1& -2023i\\ 0& 1\end{array}\right]$                     (b) $\left[\begin{array}{cc}1& 0\\ 2023i& 1\end{array}\right]$                     (c) $\left[\begin{array}{cc}1& 2023i\\ 0& 1\end{array}\right]$                    (d) $\left[\begin{array}{cc}1& 0\\ -2023i& 1\end{array}\right]$

Let A, B, C be 3 x 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) $A^{13} B^{26}-B^{26} A^{13}$ is symmetric

(S2) $A^{26} C^{13}-C^{13} A^{26}$ is symmetric

Then,

(a) Only S1 is true                                              (b) Both S1 and S2 are false

(c) Both S1 and S2 are true                               (d) Only S2 is true

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix A such that $A^2=3A+\alpha I.$ If $A^4=21A+\beta I,$ then

(a) $\beta =-8$                      (b) $\beta =8$                        (c) $\alpha =4$                       (d) $\alpha =1$

Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right).$ Then the sum of the diagonal elements of the matrix ${(A+I)}^{11}$ is equal to:

(a) 2050                               (b) 4094                               (c) 6144                                 (d) 4097

Let $A=\left[\begin{array}{ccc}0& 1& 2\\ a& 0& 3\\ 1& c& 0\end{array}\right],$ where $a,c\in \mathrm{ℝ}.$ If $A^3=A$ and the positive value of $a$ belongs to the interval $(n-1,n],$ where $n\in \mathrm{\mathbb{N}},$ then $n$ is equal to ____________ .

Let A be a symmetric matrix such that $\left|A\right|=2$ and $\left[\begin{array}{cc}2& 1\\ 3& \frac{3}{2}\end{array}\right]$ $A=\left[\begin{array}{cc}1& 2\\ \alpha & \beta \end{array}\right].$

If the sum of the diagonal elements of A is s, then $\frac{{\beta }_s}{{\alpha }^2}$ is equal to _________ .

Let $A=\left[a_{ij}\right],$ $a_{ij}\in Z\cap [0,4],$ $1\le i,$ $j\le 2.$ The number of matrices A such that the sum of all entries is a prime number $p\in (2,13)$ is _________ .

Let A be a 2 x 2 matrix with real entries such that $A^{\text{'}}=\alpha A+I,$ where $\alpha \in R-\{-1,1\}.$ If det $(A^2-A)=4,$ then the sum of all possible values of $\alpha$ is equal to

(a) 0                               (b) $\frac{3}{2}$                               (b) $\frac{5}{2}$                               (d) 2

If $\left|\begin{array}{ccc}x+1& x& x\\ x& x+\lambda & x\\ x& x& x+{\lambda }^2\end{array}\right|=\frac{9}{8}(103x+81),$ then $\lambda ,\frac{\lambda }{3}$ are the roots of the equation

(a) $4x^2-24x+27=0$                            (b) $4x^2+24x+27=0$

(c) $4x^2-24x-27=0$                            (d) $4x^2+24x-27=0$

Let $\alpha$ be a root of the equation $(a-c)x^2+(b-a)x+(c-b)=0$

where a, b, c are distinct real numbers such that the matrix $\left[\begin{array}{ccc}{\alpha }^2& \alpha & 1\\ 1& 1& 1\\ a& b& c\end{array}\right]$ is singular. Then, the value of 

$\frac{{(a-c)}^2}{(b-a)(c-b)}+\frac{{(b-a)}^2}{(a-c)(c-b)}+\frac{{(c-b)}^2}{(a-c)(b-a)}$ is

(a) 6                               (b) 3                             (c) 9                     (d) 12

The set of all values of $t\in \mathrm{ℝ}$, for which the matrix $\left[\begin{array}{ccc}{e}^t& e^{-t}(\sin t-2\cos t)& e^{-t}(-2\sin t-\cos t)\\ e^t& e^{-t}(2\sin t+\cos t)& e^{-t}(\sin t-2\cos t)\\ e^t& e^{-t}\cos t& e^{-t}\sin t\end{array}\right]$ is invertible, is

(a) $\left\{\right(2k+1)\frac{\mathrm{\pi }}{2},k\in \mathrm{\mathbb{Z}}\}$                      (b) $\mathrm{ℝ}$                      (c) $\{k\mathrm{\pi }+\frac{\mathrm{\pi }}{4},k\in \mathrm{\mathbb{Z}}\}$                       (d) $\{k\mathrm{\pi },k\in \mathrm{\mathbb{Z}}\}$

Let $D_k=\left|\begin{array}{ccc}1& 2k& 2k-1\\ n& n^2+n+2& n^2\\ n& n^2+n& n^2+n+2\end{array}\right|.$ If $\sum _{k=1}^n{D}_k=96,$ then $n$ is equal to _________ .

Let $A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 2& -1\\ 0& -1& 2\end{array}\right]$. If |adj(adj(adj 2A))| = ${\left(16\right)}^n$, then $n$ is equal to

(a) 9                             (b) 10                           (c) 12                            (d) 8

If $A=\left[\begin{array}{cc}1& 5\\ \lambda & 10\end{array}\right],$ $A^{-1}=\alpha A+\beta I$ and $\alpha +\beta =-2,$ then $4{\alpha }^2+{\beta }^2+{\lambda }^2$ is equal to

(a) 10                               (b) 14                             (c) 19                             (d) 12

If A is a $3\times 3$ matrix and |A| = 2, then $\left|3adj\right(\left|3A\right|A^2\left)\right|$ is equal to

(a) $3^{11}.6^{10}$                       (b) $3^{12}.6^{11}$                      (c) $3^{10}.6^{11}$                      (d) $3^{12}.6^{10}$

If $A=\frac{1}{5!6!7!}\left[\begin{array}{ccc}5!& 6!& 7!\\ 6!& 7!& 8!\\ 7!& 8!& 9!\end{array}\right],$ then |adj(adj(2A))| is equal to

(a) $2^{12}$                           (b) $2^{20}$                          (c) $2^8$                        (d) $2^{16}$

Let $B=\left[\begin{array}{ccc}1& 3& \alpha \\ 1& 2& 3\\ \alpha & \alpha & 4\end{array}\right],$ $\alpha >2$ be the adjoint of a matrix A and |A| = 2. Then $[\alpha -2\alpha \alpha ]$ $B\left[\begin{array}{c}\alpha \\ -2\alpha \\ \alpha \end{array}\right]$ is equal to

(a) 0                          (b) 16                         (c) 32                               (d) - 16

Let for $A=\left[\begin{array}{ccc}1& 2& 3\\ \alpha & 3& 1\\ 1& 1& 2\end{array}\right],$ |A| = 2. If |2 adj (2 adj (2A))| = ${32}^n$, then $3n+\alpha$ is equal to

(a) 10                               (b) 11                              (c) 9                           (d) 12

Let the determinant of a square matrix A of order $m$ be $m-n,$ where $m$ and $n$ satisfy $4m+n=22$ and $17m+4n=93.$ If det $(n adj (adj \left(mA\right)\left)\right)=3^a{5}^b{6}^c,$ then $a+b+c$ is equal to

(a) 109                            (b) 101                           (c) 84                           (d) 96

Let A be a $3\times 3$ matrix such that |adj(adj (adj A))| = ${12}^4.$ Then $|A^{-1}adj\hspace{0.17em}A|$ is equal to

(a) 1                            (b) $\sqrt{6}$                         (c) 12                      (d) $2\sqrt{3}$

Let $x,y,z>1$ and $A=\left[\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 2& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 3\end{array}\right].$ Then $|adj (adj A^2\left)\right|$ is equal to

(a) $4^8$                            (b) $2^8$                                 (c) $2^4$                       (d) $6^4$

Let $A=\left(\begin{array}{cc}m& n\\ p& q\end{array}\right),d=\left|A\right|\ne 0$ and |A - d(Adj A)| = 0. Then

(a) $1+d^2=m^2+q^2$                       (b) $1+d^2={(m+q)}^2$

(c) ${(1+d)}^2=m^2+q^2$                     (d) ${(1+d)}^2={(m+q)}^2$

If P is a $3\times 3$ real matrix such that $P^T=aP+(a-1)I,$ where $a>1,$ then

(a) $|Adj P|>1$                                (b) $|Adj P|=\frac{1}{2}$

(c) P is a singular matrix                      (d) $|Adj P|=1$

Let A be a $n\times n$ matrix such that |A| = 2. If the determinant of the matrix $Adj(2·Adj(2A^{-1}\left)\right)$ is $2^{84}$, then $n$ is equal to __________ .

If the system of equations

$x+y+az=b$

$2x+5y+2z=6$

$x+2y+3z=3$

has infinitely many solutions, then $2a+3b$ is equal to

(a) 20                               (b) 25                                  (c) 28                                (d) 23

For the system of equations $x+y+z=6,$ $x+2y+\alpha z=10,$ $x+3y+5z=\beta ,$ which one of the following is not true?

(a) System has infinitely many solutions for $\alpha =3,$ $\beta =14$.

(b) System has no solution for $\alpha =3,$ $\beta =24$.

(c) System has a unique solution for $\alpha =3,$ $\beta \ne 14.$

(d) System has a unique solution for $\alpha =-3,$ $\beta =14.$

Let S be the set of all values of $\theta \in [-\mathrm{\pi },\mathrm{\pi }]$ for which the system of linear equations

$x+y+\sqrt{3}z=0$

$-x+\left(\tan \theta \right)y+\sqrt{7}z=0$

$x+y+\left(\tan \theta \right)z=0$

has non-trivial solution. Then $\frac{120}{\mathrm{\pi }}\sum _{\theta \in S}\theta$ is equal to

(a) 30                              (b) 10                           (c) 40                             (d) 20

For the system of linear equations $2x-y+3z=5; 3x+2y-z=7; 4x+5y+\alpha z=\beta ,$ which of the following is NOT correct?

(a) The system has infinitely many solutions for $\alpha =-6$ and $\beta =9$

(b) The system has infinitely many solutions for $\alpha =-5$ and $\beta =9$

(c) The system has a unique solution for $\alpha \ne -5$ and $\beta =8$

(d) The system is inconsistent for $\alpha =-5$ and $\beta =8$

If the system of linear equations

$7x+11y+\alpha z=13$

$5x+4y+7z=\beta$

$175x+194y+57z=361$

has infinitely many solutions, then $\alpha +\beta +2$ is equal to:

(a) 4                              (b) 3                           (c) 6                                (d) 5

For the system of linear equations

$2x+4y+2az=b$

$x+2y+3z=4$

$2x-5y+2z=8$

which of the following is NOT correct?

(a) It has infinitely many solutions if a = 3, b = 8

(b) It has infinitely many solutions if a = 3, b = 6

(c) It has unique solution if a = b = 8

(d) It has unique solution if a = b = 6

If the system of equations $2x+y-z=5$, $2x-5y+\lambda z=\mu$, $x+2y-5z=7$ has infinitely many solutions, then ${(\lambda +\mu )}^2+{(\lambda -\mu )}^2$ is equal to

(a) 912                              (b) 916                                 (c) 904                              (d) 920

Let the system of linear equations

$-x+2y-9z=7$

$-x+3y+7z=9$

$-2x+y+5z=8$

$-3x+y+13z=\lambda$

has a unique solution $x=\alpha ,$ $y=\beta ,$ $z=\gamma$. Then the distance of the point $(\alpha ,\beta ,\gamma )$ from the plane $2x-2y+z=\lambda$ is

(a) 9                             (b) 7                              (c) 13                          (d) 11

Let S denote the set of all real values of $\lambda$ such that the system of equations

$\lambda x+y+z=1,$ $x+\lambda y+z=1,$ $x+y+\lambda z=1$ is inconsistent, then $\sum _{\lambda \in S}\left(\right|\lambda {|}^2+\left|\lambda \right|)$ is equal to

(a) 12                             (b) 4                               (c) 2                             (d) 6

For the system of linear equations $\alpha x+y+z=1,$ $x+\alpha y+z=1,$ $x+y+\alpha z=\beta ,$ which one of the following statements is NOT correct?

(a) It has infinitely many solutions if $\alpha =2$ and $\beta =-1$

(b) $x+y+z=\frac{3}{4}$ if $\alpha =2$ and $\beta =1$

(c) It has infinitely many solutions if $\alpha =1$ and $\beta =1$

(d) It has no solution if $\alpha =-2$ and $\beta =1$

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$x+y+z=1$

$2x+Ny+2z=2$

$3x+3y+Nz=3$

has unique solution is k/6, then the sum of value of k and all possible values of N is

(a) 18                              (b) 19                           (c) 20                               (d) 21

If the system of equations

$x+2y+3z=3$

$4x+3y-4z=4$

$8x+4y-\lambda z=9+\mu$

has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

(a) $(\frac{72}{5},-\frac{21}{5})$                     (b) $(\frac{72}{5},\frac{21}{5})$                      (c) $(-\frac{72}{5},-\frac{21}{5})$                        (d) $(-\frac{72}{5},\frac{21}{5})$

Let $S_1$ and $S_2$ be respectively the sets of all $a\in R-\left\{0\right\}$ for which the system of linear equations

$ax+2ay-3az=1$

$(2a+1)x+(2a+3)y+(a+1)z=2$

$(3a+5)x+(a+5)y+(a+2)z=3$

has unique solution and infinitely many solutions. Then

(a) $S_1=\Phi$ and $S_2=R-\left\{0\right\}$

(b) $S_1$ is an infinite set and $n\left(S_2\right)=2$

(c) $S_1=R-\left\{0\right\}$ and $S_2=\Phi$

(d) $n\left(S_1\right)=2$ and $S_2$ is an infinite set

Consider the following system of equations

$\alpha x+2y+z=1$

$2\alpha x+3y+z=1$

$3x+\alpha y+2z=\beta$

for some $\alpha ,\beta \in \mathrm{ℝ}.$ Then which of the following is NOT correct.

(a) It has no solution if $\alpha =-1$ and $\beta \ne 2$

(b) It has no solution for $\alpha =-1$ and for all $\beta \in \mathrm{ℝ}.$

(c) It has a solution for all $\alpha \ne -1$ and $\beta =2$

(d) It has no solution for $\alpha =3$ and for all $\beta \ne 2$

Let the system of linear equations

$x+y+kz=2$

$2x+3y-z=1$

$3x+4y+2z=k$

have infinitely many solutions. Then the system

$(k+1)x+(2k-1)y=7$

$(2k+1)x+(k+5)y=10$

has

(a) infinitely many solutions

(b) unique solution satisfying $x-y=1$

(c) no solution

(d) unique solution satisfying $x+y=1$

For $\alpha ,\beta ,\in \mathit{R},$ suppose the system of linear equations

$x-y+z=5$

$2x+2y+\alpha z=8$

$3x-y+4z=\beta$

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

(a) $x^2-18x+56=0$                     (b) $x^2+14x+24=0$

(c) $x^2-10x+16=0$                     (d) $x^2+18x+56=0$

For the system of linear equations $x+y+z=6; \alpha x+\beta y+7z=3; x+2y+3z=14,$ which of the following is NOT true?

(a) If $\alpha =\beta$ and $\alpha \ne 7,$ then the system has a unique solution

(b) There is a unique point $(\alpha ,\beta )$ on the line $x+2y+18=0$ for which the system has infinitely many solutions

(c) For every point $(\alpha ,\beta )\ne (7,7)$ on the line $x-2y+7=0,$ the system has infinitely many solutions

(d) If $\alpha =\beta =7,$ then the system has no solution

Let S be the set of values of $\lambda$, for which the system of equations

$6\lambda x-3y-3z=4{\lambda }^2,$

$2x+6\lambda y+4z=1,$

$3x+2y+3\lambda z=\lambda$ has no solution. Then $12 \underset{\lambda \in S}{\sum }\left|\lambda \right|$ is equal to ________ .