Matrix algebra

Consider the matrix
$$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$
Let $p,q,r,s,a,b,c$ and $d$ be integers such that
$$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}\quad\text{and}\quad\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$
Then which of the following statements is (are) TRUE ?

Let $\mathbb R$ denote the set of all real numbers and let $i=\sqrt{-1}$. Consider the matrices
$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$
Let $a,b,c,d$ be real numbers such that
$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$
Let
$$H=\{x+iy:\ x,y\in\mathbb R\ \text{and}\ y>0\}.$$
Then which of the following statements is (are) TRUE ?

Let $A=\begin{bmatrix}1&2\\1&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}3&3\\\beta&2\end{bmatrix}$. If $A^2-4A+I=O$ and $B^2-5B-6I=O$, then among the two statements :

(S1): $\left[(B-A)(B+A)\right]^T=\begin{bmatrix}13&15\\7&10\end{bmatrix}$

and

(S2): $\det\left(\operatorname{adj}(A+B)\right)=-5$,

Consider the matrices $A=\begin{bmatrix}2&-2\\4&-2\end{bmatrix}$ and $B=\begin{bmatrix}3&9\\1&3\end{bmatrix}$. If matrices P and Q are such that $PA=B$ and $AQ=B$, then the absolute value of the sum of the diagonal elements of $2(P+Q)$ is ________.

Let $S=\left\{A=\begin{bmatrix}a&b\\c&d\end{bmatrix}:a,b,c,d\in\{0,1,2,3,4\}\ \text{and}\ A^2-4A+3I=0\right\}$ be a set of $2\times2$ matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is:

Let M be a $3\times3$ matrix such that
$$M\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix},\quad M\begin{pmatrix}0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\1\\2\end{pmatrix}\quad\text{and}\quad M\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\1\\1\end{pmatrix}.$$
If $M\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\7\\11\end{pmatrix}$, then $x+y+z$ equals :

Let $A=\begin{bmatrix}-1&1&-1\\1&0&1\\0&0&1\end{bmatrix}$ satisfy
$$A^2+\alpha\left(\operatorname{adj}\left(\operatorname{adj}(A)\right)\right)+\beta\left(\operatorname{adj}(A)\left(\operatorname{adj}\left(\operatorname{adj}(A)\right)\right)\right)=\begin{bmatrix}2&-2&2\\-2&0&-1\\0&0&-1\end{bmatrix}$$
for some $\alpha,\beta\in\mathbb R$. Then $(\alpha-\beta)^2$ is equal to ________.

Let $A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}$ and $B=[b_{ij}]$, $1\le i,j\le3$. If $B=A^{99}-I$, then the value of $\dfrac{b_{31}-b_{21}}{b_{32}}$ is :

Let $I = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$ and $P = \begin{pmatrix} 2 & 0\\ 0 & 3 \end{pmatrix}$. Let $Q = \begin{pmatrix} x & y\\ z & 4 \end{pmatrix}$ for some non-zero real numbers $x$, $y$, and $z$, for which there is a $2\times2$ matrix $R$ with all entries being non-zero real numbers, such that $QR = RP$.

Then which of the following statements is (are) TRUE?

Let $\beta$ be a real number. Consider the matrix
$$A = \begin{pmatrix}\beta & 0 & 1\\ 2 & 1 & -2\\ 3 & 1 & -2\end{pmatrix}.$$
If $A^7 - (\beta-1)A^6 - \beta A^5$ is a singular matrix, then the value of $9\beta$ is _____.

Let $\hat i, \hat j$ and $\hat k$ be the unit vectors along the three positive coordinate axes. Let
$$\vec a = 3\hat i + \hat j - \hat k,\qquad \vec b = \hat i + b_2\hat j + b_3\hat k,\ b_2, b_3 \in \mathbb{R},\qquad \vec c = c_1\hat i + c_2\hat j + c_3\hat k,\ c_1, c_2, c_3 \in \mathbb{R}$$
be three vectors such that $b_2b_3 > 0$, $\vec a\cdot\vec b = 0$ and
$$\begin{pmatrix}0 & -c_3 & c_2\\ c_3 & 0 & -c_1\\ -c_2 & c_1 & 0\end{pmatrix}\begin{pmatrix}1\\ b_2\\ b_3\end{pmatrix} = \begin{pmatrix}3 - c_1\\ 1 - c_2\\ -1 - c_3\end{pmatrix}.$$
Then, which of the following is/are TRUE?

If $M = \begin{pmatrix}\frac52 & \frac32\\ -\frac32 & -\frac12\end{pmatrix}$, then which of the following matrices is equal to $M^{2022}$?

Let $S=\left\{\begin{pmatrix}-1&a\\0&b\end{pmatrix};a,b\in\{1,2,3,\dots,100\}\right\}$ and let $T_n=\left\{A\in S:A^{n(n+1)}=I\right\}$. Then the number of elements in $\displaystyle\bigcap_{n=1}^{100}T_n$ is ______.

Let A be a $3\times3$ real matrix such that
$$A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix};\quad A\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\\1\end{pmatrix}\quad\text{and}\quad A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}.$$
If $X=(x_1,x_2,x_3)^T$ and I is an identity matrix of order 3, then the system $(A-2I)X=\begin{pmatrix}4\\1\\1\end{pmatrix}$ has :

Let $A=\begin{bmatrix}0&-2\\2&0\end{bmatrix}$. If M and N are two matrices given by $M=\displaystyle\sum_{k=1}^{10}A^{2k}$ and $N=\displaystyle\sum_{k=1}^{10}A^{2k-1}$ then $MN^2$ is :

Let A be a $3\times3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices A having sum of all the entries equal to 5, is ______.

Let $A=\begin{pmatrix}2&-2\\1&-1\end{pmatrix}$ and $B=\begin{pmatrix}-1&2\\-1&2\end{pmatrix}$. Then the number of elements in the set $\{(n,m):n,m\in\{1,2,\dots,10\}$ and $nA^n+mB^m=I\}$ is ______.

Let A be a matrix of order $2\times2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of A is a prime number $p$, $2<p<8$, then the number of such matrices A is ______.

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

Let $A=[a_{ij}]$ be a square matrix of order 3 such that $a_{ij}=2^{j-i}$, for all $i,j=1,2,3$. Then, the matrix $A^2+A^3+\dots+A^{10}$ is equal to :