Let M and N be two 3 × 3 matrices such that M N = N M . Further, if M N 2 and M 2 = N 4 , then [2014]

Question

Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N = N M .$ Further, if $M \neq N^{2}$ and $M^{2} = N^{4},$ then [2014]

Smart Achievers · Accepted Answer

(1, 2)

$Given: M N = N M, M \neq N^{2} and M^{2} = N^{4}$

Then, $M^{2} = N^{4}$ $\Rightarrow (M + N^{2}) (M - N^{2}) = 0$

$\Rightarrow (i) M + N^{2} = 0 and M - N^{2} \neq 0,$

$(ii)$ $| M + N^{2} |$ $\neq 0 and$ $| M - N^{2} |$ $= 0$

In each case, $| M + N^{2} |$ $= 0$

$∴$ $| M^{2} + M N^{2} |$ $=$ $| M |$ $| M + N^{2} |$ $= 0$

$∴$ (1) is correct and (3) is not correct.

Also, we know if $| A |$ $= 0,$ then there can be many matrices $U$ , such that $A U = 0$

$∴$ $(M^{2} + M N^{2}) U = 0$ will be true for many values of $U$ .

$∴$ (2) is correct.

Again, if $A X = 0$ and $| A |$ $= 0,$ then $X$ can be non-zero.

$∴$ (4) is not correct.

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