Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further, if $M \neq N^{2}$ and $M^{2} = N^{4}$, then

Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N² and M² = N⁴, then $MN = NM;\quad M \ne N^2;\quad M^2 = N^4$ $M, N \text{ are } 3\times 3 \text{ matrices}$ $\text{Which of (1)--(4) is/are true?}$ $1.\ \text{factorise } M^2 - N^4$ $2.\ A = M-N^2,\ B = M+N^2,\ AB = 0$ $3.\ M^2 + MN^2 = MB \Rightarrow \det = ?$ $M^2 = N^4 \Rightarrow M^2 - N^4 = 0$ $(M-N^2)(M+N^2) = M^2 - N^4 = 0\quad (MN = NM)$ $A = M - N^2 \ne 0,\quad B = M + N^2$ $AB = 0$ $B \text{ invertible} \Rightarrow A = B^{-1}(AB) = 0\ \text{(contradiction)}$ $\Rightarrow \det B = 0$ $BA = (M+N^2)(M-N^2) = M^2 - N^4 = 0$ $M^2 + MN^2 = M(M+N^2) = MB$ $\det(M^2+MN^2) = \det(MB) = \det M \cdot \det B = 0$ $\Rightarrow (1) \text{ TRUE}$ $\det = 0 \Rightarrow \exists\, U \ne 0,\ (M^2+MN^2)U = 0$ $(M^2+MN^2)A = M(BA) = 0,\ A \ne 0 \Rightarrow (2) \text{ TRUE}$ $\det = 0 \not\ge 1 \Rightarrow (3) \text{ FALSE}$ $\text{non-zero } U \text{ exists} \Rightarrow (4) \text{ FALSE}$ $\boxed{(1)\ \text{and}\ (2)\ \text{are TRUE}}$