In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT.[1][2] Suppose A is an invertible square matrix and u, v are column vectors.
Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1.
Using unit vectors for u and/or v, individual columns, rows or elements[4] of A may be manipulated and a correspondingly updated determinant computed relatively cheaply in this way.
Suppose A is an invertible n-by-n matrix and U, V are n-by-m matrices.
Given additionally an invertible m-by-m matrix W, the relationship can also be expressed as