Let $P = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ be a matrix. Three elements of this matrix $P$ are selected at random. $A$ is the event of having the three elements whose sum is odd. $B$ is the event of selecting the three elements which are in a row or column. Then $P(A) + P(A|B) =$?

If $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$ where $\theta = \frac{2 \pi}{19}$,then $A^{2017} = $

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A^2 = 3A + \alpha I$. If $A^4 = 21A + \beta I$,then:

The total number of matrices $A = \begin{bmatrix} 0 & 2x & 2x \\ 2y & y & -y \\ 1 & -1 & 1 \end{bmatrix}$ where $x, y \in \mathbb{R}$ and $x \neq y$,for which $A^T A = 3I_3$ is:

Let $A$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}$ and $A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$. If $\det(A) = 1$ and the matrix $A$ satisfies $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$, then $\det(\text{adj}(A^2 + A))$ is equal to:

For $i=1, 2, 3$ and $j=1, 2, 3$. If $a_i^2+b_i^2+c_i^2=1$,$a_i a_j+b_i b_j+c_i c_j=0$,$\forall i \neq j$ and $A=\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$,then $\det(AA^T)=$