Let $M = [a_{i j}], i, j \in {1, 2, 3}$ be a $3 \times 3$ matrix such that $a_{i j} = 2$ if j+1 is divisible by i, other wise $a_{i j} = 0$ .
Then which of the following statements is (are) TRUE?

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The set ${X \in R^{3} : M X = 0} \neq {0}$ where $0 = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

M is invertible matrix

There exists a non–zero column matrix $[\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}]$ and such that $M [\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}] = [\begin{matrix} 4 a_{1} \\ 4 a_{2} \\ 4 a_{3} \end{matrix}]$

The matrix M+2I is not invertible, where I is the $3 \times 3$ identity matrix.

answer is B, C, D.

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Detailed Solution

$M = [\begin{matrix} 2 & 2 & 2 \\ 2 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}]$
$| M | = 0 \Rightarrow$ matrix M is not invertible matrix
$M + 2 I = [\begin{matrix} 4 & 2 & 2 \\ 2 & 2 & 2 \\ 0 & 2 & 2 \end{matrix}] \Rightarrow | M + 2 I | = 0$
$M X = 0 \Rightarrow [\begin{matrix} 2 & 2 & 2 \\ 2 & 0 & 2 \\ 0 & 2 & 0 \end{matrix}] [\begin{matrix} a_{1} \\ a_{2} \\ a_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] \Rightarrow 2 a_{1} + 2 a_{2} + 2 a_{3} = 0, 2 a_{1} + 2 a_{3} = 0, 2 a_{2} = 0 \Rightarrow a_{1} + a_{3} = 0, a_{2} = 0$
${M X = 0} \neq {0} \to$ correct
Infinite values of $a_{1}, a_{3}$ are possible
$M X = 4 X$ $\Rightarrow [M - 4 I] X = 0$
$\Rightarrow | M - 4 I | = 0$ , so solution exists for X