For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations $$\begin{array}{rl}& x-y+z=5\\ & 2x+2y+\alpha z=8\\ & 3x-y+4z=\beta \end{array}$$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :

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

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 :

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

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

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

Consider the following system of equations $$\alpha x+2y+z=1$$ $$2\alpha x+3y+z=1$$ $$3x+\alpha y+2z=\beta$$ for some $$\alpha ,\beta \in \mathbb{R}$$. Then which of the following is NOT correct.

Let A, B, C be 3 $$\times$$ 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,

Let A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right] and B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right], where $$i=\sqrt{-1}$$. If $$\mathrm{M}={\mathrm{A}}^{\mathrm{T}}\mathrm{BA}$$, then the inverse of the matrix $$\mathrm{A}{\mathrm{M}}^{2023}{\mathrm{A}}^{\mathrm{T}}$$ is

Let $$x,y,z>1$$ and A = \left[ {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 2 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 3 \cr } } \right]. Then $$|adj(\mathrm{adj}{\mathrm{A}}^2)|$$ is equal to

Let S$${}_1$$ and S$${}_2$$ be respectively the sets of all $$a\in \mathbb{R}-\{0\}$$ 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

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

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 :

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

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[ {\matrix{ {{\alpha ^2}} & \alpha & 1 \cr 1 & 1 & 1 \cr a & b & c \cr } } \right] is singular. Then, the value of {{{{(a - c)}^2}} \over {(b - a)(c - b)}} + {{{{(b - a)}^2}} \over {(a - c)(c - b)}} + {{{{(c - b)}^2}} \over {(a - c)(b - a)}} is

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

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

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

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?

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