Publish Date: June 10, 2024

Minors & Cofactors

In linear algebra, minors and cofactors are derived from a square matrix by removing one row and one column and then calculating the determinant of the remaining submatrix.

Here’s how they are defined:

Minor: The minor of an element in a matrix , denoted as , is the determinant of the matrix obtained by deleting the th row and the th column of .
Cofactor: The cofactor of an element in a matrix , denoted as , is the minor multiplied by ((-1)^{i+j}).

These concepts are often used in finding the inverse of a matrix using the adjugate matrix (also known as the adjoint matrix) and the formula for the inverse of a matrix.

Let’s say we have a matrix:

The minors and cofactors of this matrix would be:

Once we have the minors, we can calculate the cofactors by multiplying each minor by the corresponding .

These concepts are fundamental in various areas of mathematics and are particularly useful in solving systems of linear equations and finding eigenvalues and eigenvectors of matrices.