= Matrix determinant lemma
{title2=$\det(A+uv^T)=\det(A)(1+v^TA^{-1}u)$}
{wiki}
For an <invertible matrix> $A\in\mathbb R^{n\times n}$ and real <vectors> $u,v\in\mathbb R^n$, the <matrix determinant lemma> states
$$
\det(A+uv^T)=\det(A)\left(1+v^TA^{-1}u\right).
$$
The expression $v^TA^{-1}u$ is a <scalar>, so a <rank-one matrix update> changes the <determinant> by a single multiplicative factor. One need not expand the updated <determinant> entry by entry.
For the preceding example with $A=I$, the <dot product> is $v^Tu=3-2=1$. Hence the updated <determinant> is $1+1=2$. Direct evaluation confirms
$$
\det\begin{pmatrix}4&-1\\6&-1\end{pmatrix}
=4(-1)-(-1)6=2.
$$
More generally, let $B(t)=I+tuv^T$ for a real <scalar> $t$. The same calculation gives $\det B(t)=1+t$. The <determinant> is one at $t=0$, two at $t=1$, and zero at $t=-1$.
\Image[/rank-one-matrix-update-geometry.png]
{title=A rank-one correction can preserve, stretch, or collapse the area of a square}
{description=Each panel maps the unit <square> by $B(t)=I+tuv^T$ with $u=(1,2)^T$ and $v=(3,-1)^T$. Blue and orange arrows are the images of the two <standard basis> <vectors>. At $t=1$, the <parallelogram> has <area> two, despite its strong skew. At $t=-1$, the images are <linearly dependent> and the full <square> collapses onto a <line>. Panel scales differ; the stated areas come from the <determinant>, rather than from comparing pixel areas.}
A compact proof explains the scalar reduction. Put $w=A^{-1}u$ and factor $A+uv^T=A(I+wv^T)$. If $w\ne0$, choose a <basis> starting with $w$. In that <basis>, $(I+wv^T)w=(1+v^Tw)w$, while every other basis <vector> changes only by a multiple of $w$. Its <matrix> is triangular with diagonal $1+v^Tw,1,\ldots,1$, giving the required <determinant>. If $w=0$, the correction vanishes. This uses <multiplicativity of the determinant> to restore the factor $\det A$.
The denominator in the <Sherman–Morrison formula> is exactly this factor. The <rank-one update singularity criterion> therefore identifies when the updated <matrix inverse> ceases to exist. Batson, Spielman and Srivastava use both identities for the symmetric update $A+vv^T$ in https://people.eecs.berkeley.edu/~brecht/cs294docs/week4/09.Batson.sparsification.pdf[Twice-Ramanujan Sparsifiers, section 2.3, Lemmas 2.1–2.2].
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