If $A$ is a non-singular matrix such that $(A-2I)(A-3I)=O$,then $\frac{1}{5}A + \frac{6}{5}A^{-1} = $

Let $f(x) = \int \frac{7x^{10} + 9x^{8}}{(1 + x^{2} + 2x^{9})^{2}} dx$,$x > 0$,$\lim_{x \to 0} f(x) = 0$ and $f(1) = \frac{1}{4}$. If $A = \begin{bmatrix} 0 & 0 & 1 \\ \frac{1}{4} & f'(1) & 1 \\ \alpha^{2} & 4 & 1 \end{bmatrix}$ and $B = \text{adj}(\text{adj } A)$ be such that $|B| = 81$,then $\alpha^{2}$ is equal to

If $A = \begin{bmatrix} 1 & \cot \frac{\theta}{2} \\ -\cot \frac{\theta}{2} & 1 \end{bmatrix}$,then $A^{-1} =$

If $A = \begin{bmatrix} -2 & 6 \\ -5 & 7 \end{bmatrix}$,then find $adj(A)$.

Let $k$ be a positive real number and let $A = \begin{bmatrix} 2k-1 & 2\sqrt{k} & 2\sqrt{k} \\ 2\sqrt{k} & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2\sqrt{k} \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix}$. If $\det(\operatorname{adj} A) + \det(\operatorname{adj} B) = 10^6$,then $[k]$ is equal to [Note: $\operatorname{adj} M$ denotes the adjoint of a square matrix $M$ and $[k]$ denotes the greatest integer less than or equal to $k$].

If $P$ is a non-singular matrix of order $5 \times 5$ and the sum of the elements of each row is $1$,then the sum of the elements of each row in $P^{-1}$ is