Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & 0 & k \end{bmatrix}$ and $f(x) = x^3 - 2x^2 - \alpha x + \beta = 0$. If $A$ satisfies $f(A) = 0$,then:

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:

Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B = I$,where $I$ is the identity matrix and $B$ is any square matrix of the same order as $A$. The matrix product $AB$ is equal to:

If the matrix $M_r$ is given by $M_r = \begin{bmatrix} r & r-1 \\ r-1 & r \end{bmatrix}$ for $r = 1, 2, 3, \ldots$,then $\det(M_1) + \det(M_2) + \ldots + \det(M_{2008}) = $

Let $J_{n, m}=\int_{0}^{\frac{1}{2}} \frac{x^{n}}{x^{m}-1} d x, \quad \forall n>m$ and $n, m \in N$. Consider a matrix $A=\left[a_{i j}\right]_{3 \times 3}$ where $a_{i j}=J_{6+i, 3}-J_{i+3,3}$ for $i \leq j$ and $a_{i j}=0$ for $i>j$. Then $\left|\operatorname{adj} A^{-1}\right|$ is:

If $A$ and $B$ are square matrices of the same order such that $AB = A$ and $BA = B$,then $(A + I)^5$ is equal to (where $I$ is the identity matrix).