Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ and $B=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that $A B=B$ and $a+b=2021$, then the value of...

Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew...

Let $I$ be an identity matrix of order $2 \times 2$ and $P=\left[\begin{array}{ll}2 & -1 \\ 5 & -3\end{array}\right]$. Then the value of $...

If for the matrix, $A=\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right], A A^{\top}=I_2$, then the value of $\alpha^4+\beta^4$ is :

Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $\alpha \in R$. Suppose $...

Let $A=\left[\begin{array}{l}a_1 \\ a_2\end{array}\right]$ and $B=\left[\begin{array}{l}b_1 \\ b_2\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $\mathrm{A}=\mathrm{XB}$, where $...

Let $\mathrm{A}=\left\{\mathrm{a}_{\mathrm{ij}}\right\}$ be a $3 \times 3$ matrix, where $...

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the...

If $A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)$ and $\operatorname{det}=\left(A^2-\frac{1}{2} I\right)=0$ then a possible value of $\alpha$ is

For the system of linear equations : $$ x-2 y=I, x-y+k z=-2, k y+4 z=6, k \in R, $$ consider the following statements: (A) The...