Matrix determinant lemma

The Matrix Determinant Lemma provides a way to calculate how adding a rank-one update (a simple type of change) to a square matrix affects its determinant (a value that summarizes certain properties of the matrix). Specifically, if you have a matrix \(A\) and vectors \(u\) and \(v\), then the determinant of \(A + uv^T\) can be computed using the original determinant of \(A\) and the vectors \(u\) and \(v\). This lemma simplifies the process of understanding how small, specific modifications impact overall matrix properties without recalculating the entire determinant.