For matrix $A$, if $A^4 = 0$ does this also mean that $A^2 = 0$?

Question

For an arbitrary matrix $A$, if $A^4 = 0$ does this also mean that $A^2 = 0$?

My thinking is that it does since I can reduce $A^4$ into $(A^2)^2$ but I'm not sure if this helps or not.

Answer 1

$$\pmatrix{0&1&0\\0&0&1\\0&0&0}$$

Answer 2

Another counterexample, when the matrix elements are taken from $\mathbb Z/4\mathbb Z$, which is a commutative ring with a zero divisor: $$ A=\pmatrix{1&-1\\ 1&1},\ A^2=\pmatrix{0&-2\\ 2&0},\ A^4=-4I=0. $$ But surely, one can construct even an $1\times1$ counterexample, such as $A=2$ over $\mathbb Z/2^4\mathbb Z$.