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 $A = \begin{bmatrix} \cos^2 \alpha & \sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \sin \beta \cos \beta \\ \sin \beta \cos \beta & \sin^2 \beta \end{bmatrix}$ are such that $AB$ is a null matrix,then which of the following must be an odd integral multiple of $\frac{\pi}{2}$?

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

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 $n \in N$.

If $A$ and $B$ are both $3 \times 3$ matrices,then which of the following statements are true?
$(i)$ $AB=0 \Rightarrow A=0$ or $B=0$
(ii) $AB=I_3 \Rightarrow A^{-1}=B$
(iii) $(A-B)^2=A^2-2AB+B^2$

Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is: