= Invertibility of identity plus a real skew-symmetric matrix
{title2=$A^\top=-A\Rightarrow I+A\text{ invertible}$}
\b[Proof.] If $(I+A)x=0$, the <quadratic-form criterion for a skew-symmetric matrix> gives $0=x^\top(I+A)x=\|x\|^2$, so $x=0$. The <rank-nullity theorem> makes the square <matrix> $I+A$ invertible. More generally $tI+A$ is invertible for every nonzero real $t$, since its <null space> would imply $t\|x\|^2=0$. Equivalently, the only possible real <eigenvalue> of a real <skew-symmetric matrix> is zero.
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