Special matrices

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.

Consider the matrix

$$P = \begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{pmatrix}.$$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times3$ invertible matrices $Q$ with integer entries, such that

$$Q^{-1} = Q^T \quad\text{and}\quad PQ = QP,$$

is

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3\times 3$ matrix $M = (a_{ij})_{3\times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

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 $A=\begin{pmatrix}2&-2\\1&-1\end{pmatrix}$ 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 $X=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}$, $Y=\alpha I+\beta X+\gamma X^2$ and $Z=\alpha^2I-\alpha\beta X+\left(\beta^2-\alpha\gamma\right)X^2$, $\alpha,\beta,\gamma\in\mathbb R$. If $Y^{-1}=\begin{bmatrix}\frac15&-\frac25&\frac15\\0&\frac15&-\frac25\\0&0&\frac15\end{bmatrix}$, then $(\alpha-\beta+\gamma)^2$ is equal to ______.

The number of matrices $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d\in\{-1,0,1,2,3,\ldots,10\}$, such that $A=A^{-1}$, is ______.

Let S be the set containing all $3\times3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{\mathrm T}A$ is 6 is ______.

Let the matrix $A=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\operatorname{Adj}(B_{n-1})$ for all $n\ge1$, then $\det(B_4)$ is equal to :

Let A and B be any two $3\times3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let

$$P_1 = I = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix},\quad P_2 = \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix},\quad P_3 = \begin{bmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix},$$

$$P_4 = \begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix},\quad P_5 = \begin{bmatrix}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix},\quad P_6 = \begin{bmatrix}0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{bmatrix}$$

and

$$X = \sum_{k=1}^6P_k\begin{bmatrix}2 & 1 & 3\\ 1 & 0 & 2\\ 3 & 2 & 1\end{bmatrix}P_k^T$$

where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?

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 :

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 $X$ and $Y$ be two arbitrary, $3\times3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

If $A=\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}$ is a matrix satisfying the equation $AA^T=9I$, where I is 3×3 identity matrix, then the ordered pair (a, b) is equal to :

Let $M$ be a $2\times2$ symmetric matrix with integer entries. Then $M$ is invertible if

If A is an $3\times3$ non - singular matrix such that $AA'=A'A$ and $B=A^{-1}A'$, then $BB'$ equals :

Let A and B be any two 3×3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB − BA is :

For $3\times3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?

Let $M$ and $N$ be two $3\times3$ non-singular skew-symmetric matrices such that $MN=NM$. If $P^T$ denotes the transpose of $P$, then $M^2N^2\left(M^TN\right)^{-1}\left(MN^{-1}\right)^T$ is equal to