Let $A = \begin{bmatrix} 1 & 2 \\ -2 & -5 \end{bmatrix}$. Let $\alpha, \beta \in \mathbb{R}$ be such that $\alpha A^{2} + \beta A = 2I$. Then $\alpha + \beta$ is equal to -

Suppose $a_1, a_2, \dots$ are real numbers,with $a_1 \neq 0$. If $a_1, a_2, a_3, \dots$ are in $A.P.$,then:

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 $n \in N$.

Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

If $\Delta_1=\left|\begin{array}{lll}1 & a^2 & a^3 \\ 1 & b^2 & b^3 \\ 1 & c^2 & c^3\end{array}\right|$ and $\Delta_2=\left|\begin{array}{lll}b c & b+c & 1 \\ c a & c+a & 1 \\ a b & a+b & 1\end{array}\right|$, then $\frac{\Delta_1}{\Delta_2}=$

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: