Matrices and Determinants

Which one of the following matrices can be obtained by performing elementary row transformations on the $3\times3$ identity matrix ?

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 ?

For real numbers $\alpha,\ \beta,\ \gamma,\ \delta$ and $\mu$, consider the matrix
$$M=\begin{bmatrix}\alpha&\frac1{\sqrt2}&-\frac1{\sqrt2}\\\frac1{\sqrt3}&\beta&\frac1{\sqrt3}\\\gamma&\delta&\mu\end{bmatrix}.$$
Suppose that $MM^T=I$, where $M^T$ is the transpose of the matrix $M$, and $I$ is the $3\times3$ identity matrix. Let
$$\vec u=\alpha\,\hat i+\frac1{\sqrt3}\hat j+\gamma\,\hat k,\qquad\vec v=\frac1{\sqrt2}\hat i+\beta\hat j+\delta\hat k\quad\text{and}\quad\vec w=-\frac1{\sqrt2}\hat i+\frac1{\sqrt3}\hat j+\mu\hat k.$$
Match each entry in List-I to the correct entry in List-II and choose the correct option.

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 $\alpha,\beta\in\mathbb R$ be such that the system of linear equations
$$x+2y+z=5,\qquad 2x+y+\alpha z=5,\qquad 8x+4y+\beta z=18$$
has no solution. Then $\dfrac\beta\alpha$ is equal to :

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

If the system of equations
$$x+5y+6z=4,\qquad 2x+3y+4z=7,\qquad x+6y+az=b$$
has infinitely many solutions, then the point $(a,b)$ lies on the line

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 $A=\begin{bmatrix}1&1&2\\-2&0&1\\1&3&5\end{bmatrix}$. Then the sum of all elements of the matrix $\operatorname{adj}\left(\operatorname{adj}\left(2(\operatorname{adj}A)^{-1}\right)\right)$ is equal to:

If the system of equations:
$$x+y+z=5,\qquad x+2y+3z=9,\qquad x+3y+\lambda z=\mu$$
has infinitely many solutions, then the value of $\lambda+\mu$ is:

Let $A=\begin{bmatrix}1&2&7\\4&-2&8\\3&8&-7\end{bmatrix}$ and $\det(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2p)^2=320$, intersects the co-ordinate axes at

Let A be a $3\times3$ matrix such that
$$A^T\begin{bmatrix}1\\0\\1\end{bmatrix}=\begin{bmatrix}5\\2\\2\end{bmatrix},\quad A^T\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}3\\1\\1\end{bmatrix},\quad A\begin{bmatrix}1\\0\\1\end{bmatrix}=\begin{bmatrix}3\\4\\4\end{bmatrix}\quad\text{and}\quad A\begin{bmatrix}0\\0\\1\end{bmatrix}=\begin{bmatrix}1\\3\\1\end{bmatrix}.$$
If $\det(A)=1$, then $\det\left(\operatorname{adj}(A^2+A)\right)$ is equal to:

Consider the system of linear equations in $x,y,z$:
$$x+2y+tz=0,\qquad 6x+y+5tz=0,\qquad 3x+t^2y+f(t)z=0,$$
where $f:\mathbb R\to\mathbb R$ is a differentiable function. If this system has infinitely many solutions for all $t\in\mathbb R$, then $f$

If $f:\mathbf N\to\mathbf Z$ is defined by
$$f(n)=\begin{vmatrix}n&-1&-5\\-2n^2&3(2k+1)&2k+1\\-3n^3&3k(2k+1)&3k(k+2)+1\end{vmatrix},\quad k\in\mathbf N,$$
and $\displaystyle\sum_{n=1}^{k}f(n)=98$, then k is equal to :

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 ________.

The sum of all possible values of $\theta\in[0,2\pi]$, for which the system of equations :
$$x\cos3\theta-8y-12z=0,\qquad x\cos2\theta+3y+3z=0,\qquad x+y+3z=0$$
has a non-trivial solution, 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 $A=\begin{bmatrix}1&3&-1\\2&1&\alpha\\0&1&-1\end{bmatrix}$ be a singular matrix. Let $f(x)=\displaystyle\int_0^x\left(t^2+2t+3\right)dt$, $x\in[1,\alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1,\alpha]$, then $3(M-m)$ is equal to :