Square-zero criterion for a two-by-two matrix ($A^2=0$)

For a real or complex two-by-two matrix $A$, the Cayley-Hamilton theorem reads $A^2-(\mathrm{tr}A)A+(\det A)I=0$. Hence $A^2=0$ exactly when $\mathrm{tr}A=\det A=0$. Conversely, if $A^m=0$ for any positive integer $m$, every eigenvalue is zero and the same identity gives $A^2=0$. Thus a nonzero nilpotent matrix of this size has nilpotency index two.