Advanced Linear Algebra, Lecture 3.6: Minors and cofactors

Advanced Linear Algebra, Lecture 3.6: Minors and cofactors

The most common algorithm for computing determinants involves crossing out the i'th row and j'th column to obtain an (n-1)x(n-1) submatrix A_{ij}. The (i,j) minor is the determinant of this matrix and the (i,j) cofactor is this times (-1)^{i+j}. In this lecture, we derive the popular Laplace expansion, which says that det(A) is the linear combination of cofactors by the coefficients, down any column or across any row. We also see how Cramer's rule gives a formula for the solution of a system Ax=b in terms of cofactors and det(A). This also gives us a simple formula for the inverse of a matrix. Unfortunately, these formulas are not practical because computing determinants is a computationally expensive task. However, we will need these formulas for proofs later on.

[cascave]

thank you! cramer's rule makes so much more sense now