Minors and Cofactors

Minors and Cofactors A minor is a submatrix of a matrix that is removed from the original matrix. The size of the minor is equal to the size of the original...

Minors and Cofactors

A minor is a submatrix of a matrix that is removed from the original matrix. The size of the minor is equal to the size of the original matrix minus the number of rows or columns removed.

Cofactor is the scalar value that results from deleting the row and column of the minor.

For example, the minor formed by removing the first row and first column of the matrix

$\begin{matrix} 1 & 2 \\\ 3 & 4 \\\ 5 & 6 \end{matrix}$

would be the following submatrix:

$\begin{matrix} 2 \\\ 4 \\\ 6 \end{matrix}$

The cofactor of the minor would be 2.

The sum of the minors of a matrix is equal to the determinant of the original matrix. This means that the determinant of the original matrix can be calculated by summing the determinants of all the minors.

$\det(A) = \det(\begin{matrix} 1 & 2 \\\ 3 & 4 \\\ 5 & 6 \end{matrix}) = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2$

The determinant of the original matrix is -2, which is equal to the sum of the minors of the matrix