IPMAT Indore 2025 (SA) PYQs

Let $A, B, C$ be three $4 \times 4$ matrices such that $det \ A = 5, det \ B = -3$, and $det \ C = \frac{1}{2}$. Then the $det$ $2AB^{-1}C^3B^T$ is

If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$, then the absolute value of the determinant of $\left(A^{9}+A^{6}+A^{3}+A\right)$ is __________.

A $2 \times 2$ matrix is filled with four distinct integers randomly chosen from the set $\{1,2,3,4,5,6\}$. Then the probability that the matrix generated in such a way is singular is