Let $A$ be a $3 \times 3$ matrix such that $A^T + 2A = I$. Then $\det(A^{-1})$ is equal to:

If $A$ and $B$ are square matrices of the same order such that $(A+B)(A-B)=A^{2}-B^{2}$,then $(A B A^{-1})^{2}$ is equal to

Matrix $A = \begin{bmatrix} x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z \end{bmatrix}$. If $xyz = 60$ and $8x + 4y + 3z = 20$,then $A (adj A)$ is equal to:

If matrix $A = \begin{bmatrix} 1 & 0 & -1 \\ 3 & 4 & 5 \\ 0 & 6 & 7 \end{bmatrix}$ and its inverse is denoted by $A^{-1} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$,then the value of $a_{23}$ is:

Let $A = \left[ {\begin{array}{*{20}{c}}1&0&0\\2&1&0\\3&2&1\end{array}} \right]$. If $u_1$ and $u_2$ are column matrices such that $A{u_1} = \left[ {\begin{array}{*{20}{c}}1\\0\\0\end{array}} \right]$ and $A{u_2} = \left[ {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right]$,then $u_1 + u_2$ is equal to:

If $A = \begin{bmatrix} a & b & c \\ d & e & f \\ l & m & n \end{bmatrix}$ is a matrix such that $|A| > 0$ and $\text{Adj}(A) = \begin{bmatrix} 0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4 \end{bmatrix}$,then $\frac{cd}{fb} + \frac{\ln}{em} = $