Given $A=\left(\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right)$ and $f(x)=x^2-2 x-3$ then $f(A)$ is:

Given $P=\left[\begin{array}{lll}2 & \boldsymbol{\alpha} & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 3\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix A and $|A|=3$, then the value of $\boldsymbol{\alpha}$ is:

Given the matrices $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{lll}2 & 1 & 0 \\ 1 & 1 & 2 \\ 0 & 2 & 1\end{array}\right]$, then the minor $\boldsymbol{M}_{\mathbf{2 3}}$ of the matrix $\left(A B^{-1}\right)^{-1}$ is:

Kiran purchased 3 pencils, 2 notebooks and one pen for ₹41. From the same shop Manasa purchased 2 pencils, one notebook and 2 pens for ₹ 29 , while Shreya purchased 3 pencils, 2 notebooks and 2 pens for ₹ 44. The above situation can be represented in matrix form as $A X=B$. Then $|\operatorname{adj} A|$ is equal to