Embibe Experts Solutions for Chapter: Matrices and Determinants, Exercise 2: Exercise-2

$A$ and $B$ are two matrices such that $AB=B, BA=A$ then $A^2+B^2=$

If $P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$ and $Q=PA{P}^T$ and $X=P^T{Q}^{2005}P,$ then $X$ is equal to

The value of $a$ for which system of equation $a^3\mathrm{x}+(a+1{)}^3\mathrm{y}+(a+2{)}^3\mathrm{z}=0$, $ax+\left(a+1\right)y+\left(a+2\right)z=0$ and $x+y+z=0$ has non-zero solutions is

Let $A$ and $B$ be two $2\times 2$ matrix with real entries. If $AB=O$ and $tr\left(A\right)=tr\left(B\right)=0$ then

If $A^{-1}=\left[\begin{array}{ccc}1& -1& 0\\ 0& -2& 1\\ 0& 0& -1\end{array}\right],$ then

If $A$ and $B$ are the square matrices of order $3,$ then the true statement is$/$are (where $I$ is unit matrix).

Let $M$ be a $3\times 3$ non-singular matrix with $det\left(M\right)=4.$ If $M^{-1}\mathrm{adj}(\mathrm{adj} M)=k^{2}I,$ then the value of  $\text{'}k\text{'}$ may be:

If $AX=B,$ where $A$ is $3\times 3$ and $X$ and $B$ are $3\times 1$ matrices, then which of the following is correct?