JEE Advanced 2023 Paper 2Question 1
Let $M=(a_{ij}),\ i,j\in\{1,2,3\}$, be the $3\times3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following statements is(are) true?
Matrices
Solution
$M=\begin{pmatrix}1&1&1\\1&0&1\\0&1&0\end{pmatrix}$, $\det M=0$ so (A) false, (C) true. $\det(M+I)=0$ so $MX=-X$ has a nonzero solution: (B) true. $\det(M-2I)=0$ so (D) false.