Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{bmatrix}$. Then the sum of the diagonal elements of the matrix $(A + I)^{11}$ is equal to:

Let $A=[a_{ij}]$ be a matrix of order $3 \times 3$,with $a_{ij}=(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha+\beta \sqrt{2}$,where $\alpha, \beta \in Z$,then $\alpha+\beta$ is equal to

If $A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$,prove that $A^{n}=\begin{bmatrix} 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \\ 3^{n-1} & 3^{n-1} & 3^{n-1} \end{bmatrix}$ for all $n \in N$.

Let $a_1, a_2, a_3, \dots, a_{10}$ be in $G.P.$ with $a_i > 0$ for $i = 1, 2, \dots, 10$ and $S$ be the set of pairs $(r, k)$,$r, k \in N$ (the set of natural numbers) for which
$\left| \begin{array}{ccc} \log_e(a_1^r a_2^k) & \log_e(a_2^r a_3^k) & \log_e(a_3^r a_4^k) \\ \log_e(a_4^r a_5^k) & \log_e(a_5^r a_6^k) & \log_e(a_6^r a_7^k) \\ \log_e(a_7^r a_8^k) & \log_e(a_8^r a_9^k) & \log_e(a_9^r a_{10}^k) \end{array} \right| = 0$
Then the number of elements in $S$ is:

Let $p, q, r$ be three real numbers satisfying $[p \, q \, r] \begin{bmatrix} 2 & p & q \\ -3 & q & -p+r \\ 12 & r & -q+3r \end{bmatrix} = [5 \, b \, c]$. Then the minimum value of $(b+c)$ is:

If $A = \begin{bmatrix} 1 + a^2 + a^4 & 1 + ab + a^2b^2 & 1 + ac + a^2c^2 \\ 1 + ab + a^2b^2 & 1 + b^2 + b^4 & 1 + bc + b^2c^2 \\ 1 + ac + a^2c^2 & 1 + bc + b^2c^2 & 1 + c^2 + c^4 \end{bmatrix}$ and $\det(A) = \det(4I)$,where $I$ is a $3 \times 3$ identity matrix,then $(a - b)^3 + (b - c)^3 + (c - a)^3$ can be equal to -