Let $M = (a_{ij})$, $i, j \in \{1, 2, 3\}$, be the $3 \times 3$ 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?

Let M = (aᵢⱼ), i, j ∈ {1, 2, 3}, be the 3 × 3 matrix such that aᵢⱼ = 1 if j + 1 is divisible by i, otherwise aᵢⱼ = 0. Then which of the following statements is(are) true?

$a_{ij} = 1 \text{ if } i \mid (j+1), \text{ else } a_{ij} = 0$

$i, j \in \{1, 2, 3\}$

$\text{To find: which of the statements (1)–(4) are true}$

$a_{ij} = 1 \iff i \mid (j+1), \quad j+1 \in \{2,3,4\}$

M = 1 1 1 1 0 1 0 1 0

$\det M = 1(0-1) - 1(0-0) + 1(1-0) = -1 + 1 = 0$

$M \text{ is singular} \Rightarrow (1)\ \text{is false}$

$\det M = 0 \Rightarrow \exists\, X \neq 0 \text{ with } MX = 0$

$\{X : MX=0\} \neq \{0\} \Rightarrow (3)\ \text{is true}$

$Mx = -x \iff (M+I)x = 0$

M+I = 2 1 1 1 1 1 0 1 1

$\det(M+I) = 2(1-1)-1(1-0)+1(1-0) = 0 \Rightarrow x=(0,1,-1) \neq 0;\ (2)\ \text{is true}$

M-2I = -1 1 1 1 -2 1 0 1 -2

$\det(M-2I) = -1(4-1)-1(-2-0)+1(1-0) = 0 \Rightarrow (4)\ \text{is false}$

$\boxed{\text{True statements: (2) and (3)}}$