If $P$ and $Q$ are two non-zero square matrices of the same order such that the product $PQ = 0$,then ........

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:

$A = [a_{ij}]$ is a $3 \times 3$ matrix with positive integers as its elements. Elements of $A$ are such that the sum of all the elements of each row is equal to $6$ and $a_{22} = 2$. If $a_{ii} = \begin{cases} a_{ij} + a_{ji}, & j = i + 1 \text{ when } i < 3 \\ a_{ij} + a_{ji}, & j = 4 - i \text{ when } i = 3 \end{cases}$ for $i = 1, 2, 3$, then $|A| = $

Let $A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \text{adj}(A)$. If $\det(B) = 66$, then $\det(\text{adj}(A))$ equals:

If $A+B=\left[\begin{array}{cr}1 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 1\end{array}\right]$ where $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix,then the matrix $\left(A^{-1} B+A B^{-1}\right)$ at $\theta=\frac{\pi}{6}$ is given by

Let $A$ and $B$ be two symmetric matrices of order $3$.
Statement $-1$: $A(BA)$ and $(AB)A$ are symmetric matrices.
Statement $-2$: $AB$ is a symmetric matrix if the matrix multiplication of $A$ with $B$ is commutative.