Let $A = [a_{ij}]$ be a $3 \times 3$ matrix,where
$a_{ij} = 1$,if $i = j$
$a_{ij} = -x$,if $|i - j| = 1$
$a_{ij} = 2x + 1$,otherwise
Let a function $f: R \rightarrow R$ be defined as $f(x) = \det(A)$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:

If $\Delta_{r}=\left|\begin{array}{cc}\frac{1}{3r-2} & \frac{2}{3r-5} \\ 0 & \frac{3}{3r+1}\end{array}\right|$, then $\sum_{r=1}^{33} \Delta_{r}=$

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \text{adj}(A)$. If $\det(B) = 66$, then $\det(\text{adj}(A))$ equals:

Let $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $P = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$. Let $Q = \begin{bmatrix} x & y \\ z & 4 \end{bmatrix}$ for some non-zero real numbers $x, y$,and $z$,for which there exists a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers,such that $QR = RP$. Then which of the following statements is (are) true?

The total number of $3 \times 3$ matrices $A$ having entries from the set $\{0, 1, 2, 3\}$ such that the sum of all the diagonal entries of $AA^{T}$ is $9$,is equal to........

Let $A$ be a $2 \times 2$ matrix with real entries such that $A^{T} = \alpha A + I$,where $\alpha \in R - \{-1, 1\}$. If $\det(A^2 - A) = 4$,then the sum of all possible values of $\alpha$ is equal to