Is it possible to find two matrices

Question

Is it possible to find matrices $M$ and $N$ such that

$MN=O$ and $NM=I$

where $0=$ the $3\times3$ zero matrix and $I=$ the $2\times 2$ Identity matrix

Answer 1

Hint: $\text{trace}(MN)=\text{trace}(NM)$.

Answer 2

Suppose it were possible.

We have that

$$tr(MN) = tr(O) = 0$$

and

$$tr(NM) = tr(I) = 2$$

but since

$$tr(MN) = tr(NM)$$

we must have that

$$0 = tr(MN) = tr(NM) = 2,$$

which is absurd.