Matrix algebra

For two 3 × 3 matrices A and B, let $A+B=2B'$ and $3A+2B=I_3$, where B′ is the transpose of B and $I_3$ is 3 × 3 identity matrix. Then :

Let $z=\dfrac{-1+\sqrt3\,i}{2}$, where $i=\sqrt{-1}$, and $r,s\in\{1,2,3\}$. Let $P=\begin{bmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{bmatrix}$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is

Let $P=\begin{bmatrix}1&0&0\\4&1&0\\16&4&1\end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$, then $\dfrac{q_{31}+q_{32}}{q_{21}}$ equals

If $A=\begin{bmatrix}5a&-b\\3&2\end{bmatrix}$ and $A\operatorname{adj}A=AA^T$, then $5a+b$ is equal to :

If $P=\begin{bmatrix}\frac{\sqrt3}2&\frac12\\-\frac12&\frac{\sqrt3}2\end{bmatrix}$, $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q=PAP^T$, then $P^TQ^{2015}P$ is :

Let A be a 3 × 3 matrix such that $A^2-5A+7I=O$.

Statement - I : $A^{-1}=\dfrac17(5I-A)$.

Statement - II : The polynomial $A^3-2A^2-3A+I$ can be reduced to 5(A − 4I).

Then :

If $A=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$, then which one of the following statements is not correct ?

Let $M$ and $N$ be two $3\times3$ matrices such that $MN=NM$. Further, if $M\ne N^2$ and $M^2=N^4$, then

If B is a $3\times3$ matrix such that $B^2=0$, then $\det\left[(I+B)^{50}-50B\right]$ is equal to :

Let A be a $3\times3$ matrix such that
$$A\begin{bmatrix}1&2&3\\0&2&3\\0&1&1\end{bmatrix}=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}.$$
Then $A^{-1}$ is :

If $A=\begin{bmatrix}1&2&x\\3&-1&2\end{bmatrix}$ and $B=\begin{bmatrix}y\\x\\1\end{bmatrix}$ be such that $AB=\begin{bmatrix}6\\8\end{bmatrix}$, then :

Let $\omega$ be a complex cube root of unity with $\omega\ne1$ and $P=[p_{ij}]$ be a $n\times n$ matrix with $p_{ij}=\omega^{i+j}$. Then $P^2\ne0$, when $n=$

If $P$ is a $3\times3$ matrix such that $P^T=2P+I$, where $P^T$ is the transpose of $P$ and $I$ is the $3\times3$ identity matrix, then there exists a column matrix $X=\begin{bmatrix}x\\y\\z\end{bmatrix}\ne\begin{bmatrix}0\\0\\0\end{bmatrix}$ such that

Let $M$ be a $3\times3$ matrix satisfying
$$M\begin{bmatrix}0\\1\\0\end{bmatrix}=\begin{bmatrix}-1\\2\\3\end{bmatrix},\quad M\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}1\\1\\-1\end{bmatrix},\quad\text{and}\quad M\begin{bmatrix}1\\1\\1\end{bmatrix}=\begin{bmatrix}0\\0\\12\end{bmatrix}.$$
Then the sum of the diagonal entries of $M$ is