The number of values of $\theta \in(0, \pi)$ for which the system of linear equations $$ \begin{aligned} & x+3 y+7 z=0 \\ & -x+4 y+7...

If the matrix $A=\left[\begin{array}{cc}0 & 2 \\ k & -1\end{array}\right]$ satisfies $A\left(A^3+3 I\right)=2 I$, then the value of $K$ is :

If $A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}$, and $Q=A^{\top} B A$, then the inverse of the...

Let $A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)$, where $[t]$ denotes the greatest integer less...

Let A be a $3 \times 3$ real matrix. If $\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 A))))=2^{41}$, then the value of $\operatorname{det}\left(A^2\right)$ equal $\_\_\_\_$ .

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$, then the matrix $A^{-50}$ when $\theta=\frac{\pi}{12}$, is equal to :

Let $\quad \mathrm{A}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right),(\alpha \in \mathrm{R}) \quad$ such that $A^{32}=\left(\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right)$....

The greatest value of $c \in R$ for which the system of linear equations $$ \begin{aligned} & x-c y-c z=0 \\ & c x-y+c z=0...

The system of linear equations $$ \begin{aligned} & x+y+z=2 \\ & 2 x+3 y+2 z=5 \\ & 2 x+3 y+\left(a^2-1\right) z=a+1 \end{aligned} $$

Let $\mathrm{A}=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{array}\right)$. Then $\mathrm{A}^{2025}-\mathrm{A}^{2020}$ is equal to :