Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

If $A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & -4 \\ -3 & r & s \end{bmatrix}$ is a skew-symmetric matrix,then $|A| + |B| - |AB| = $

Let $p$ and $p+2$ be prime numbers and let $\Delta=\left|\begin{array}{ccc}p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)!\end{array}\right|$. Then the sum of the maximum values of $\alpha$ and $\beta$,such that $p^{\alpha}$ and $(p+2)^{\beta}$ divide $\Delta$,is $........$

For a square matrix $B$ of order $3$, if $B^T=B^{-1}$ and $|B|=1$, then $|B-I|=$

Let $P$ and $Q$ be $3 \times 3$ matrices such that $P \neq Q$. If $P^3 = Q^3$ and $P^2Q = Q^2P$,then the determinant $\det(P^2 + Q^2)$ is equal to:

Let $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $A$ be a $2 \times 2$ matrix satisfying $\left(A^T\right)^{-1}=A$. If $X=A B A^T$, then $A^T X^{2021} A=$