Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum...

Let $M=\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right]$, where $\alpha$ is a non-zero real number an $N=\sum_{k=1}^{49} M^{2 k}$. If $\left(I-M^2\right) N=-2 I$, then the...

Let $A=\left(\begin{array}{cc}1+i & 1 \\ -i & 0\end{array}\right)$ where $i=\sqrt{-1}$. Then, the number of elements in the set $\left\{n \in\{1,2, \ldots, 100\}: A^n=A\right\}$ is

Let $X=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right], Y=\alpha l+\beta x+\gamma x^2$ and $...

Let $\mathrm{S}=\left\{\left(\begin{array}{cc}-1 & \mathrm{a} \\ 0 & \mathrm{~b}\end{array}\right) ; \mathrm{a}, \mathrm{b} \in\{1,2,3, \ldots 100\}\right\}$ and let $\mathrm{T}_{\mathrm{n}}=\left\{\mathrm{A} \in \mathrm{S}: \mathrm{A}^{\mathrm{n}(\mathrm{n}+1)}=\mathrm{I}\right\}$. Then the number of elements...

Let $A=\left(\begin{array}{ll}2 & -2 \\ 1 & -1\end{array}\right)$ and $B=\left(\begin{array}{ll}-1 & 2 \\ -1 & 2\end{array}\right)$. Then the number of elements in the set $...

Let $A$ and $B$ be two $3 \times 3$ matrices such that $A B=I$ and $|A|=\frac{1}{8}$ then $|\operatorname{adj}(\operatorname{Badj}(2 A))|$ is equal to

Let $\mathrm{A}=\left[a_{i j}\right]$ be a square matrix of order 3 such that $a_{i j}=2^{i-i}$, for all $\mathrm{i}_{\text {and }} \mathrm{j}=1,2,3$. Then, the matrix $\mathrm{A}^2+\mathrm{A}^3+\ldots+ \mathrm{A}^{10}$...

The positive value of the determinant of the matrix $A$, whose $...