= Powers of the identity plus a square-zero matrix
{title2=$(I+N)^k=I+kN,\quad N^2=0,\quad k\in\mathbb Z$}
The <binomial theorem> for the commuting <matrices> $I,N$ proves the formula for nonnegative integers: every term containing $N^2$ vanishes. Since $(I+N)(I-N)=I$, the <matrix inverse> is $I-N$; applying the same argument to its powers gives the negative integers as well. The same identity shows that $I+N$ is invertible even when the <nilpotent matrix> $N$ is nonzero. In dimension two, the example $N=\left(\begin{smallmatrix}0&p\\0&0\end{smallmatrix}\right)$ describes repeated <shear mappings>, whose displacement parameters add under composition.
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