Square-zero criterion for a two-by-two matrix
= Square-zero criterion for a two-by-two matrix
{title2=$A^2=0$}
For a real or complex two-by-two <matrix> $A$, the <Cayley-Hamilton theorem> reads $A^2-(\operatorname{tr}A)A+(\det A)I=0$. Hence $A^2=0$ exactly when $\operatorname{tr}A=\det A=0$. 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.