Alternative proofs Matrix Determinant Lemma

Question

Well as many of you know wiki has a beautiful proof for the Matrix Determinant Lemma Wiki's Proof

But: How the hell is one supposed to get there on his own? There is no way that when a professor would ask you to proof the lemma in your let's say 4th semester, you will come up with the idea for that proof. So my question is: Are there alternative methods to show the Lemma which are more intuitiv or let's say realistic to come up with?

Answer 1

Start with the lower triangular matrix $I+ue_1^T$ where you know the determinant to be $1+u_1=1+e_1^Tu$.

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Next, you know that there exist matrices that reflect or rotate any vector $v$ into a multiple of $e_1$, $Qv=\alpha e_1\iff v=αQ^Te_1$ (Householder reflectors, basis completion to an orthonormal basis,...).

Then $$ \det(I+uv^T)=\det(I+αue_1^TQ)=\det(I+αQue_1^T)=1+αe_1^TQu=1+v^Tu. $$

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After that the general case is simple, $$ \det(A+uv^T)=\det(A)\det(I+A^{-1}uv^T)=\det(A)(1+v^TA^{-1}u). $$