Minor (Linear Algebra)

linear-algebra

Definition

Minor

A minor $M_{ij}$ of an element $a_{ij}$ of a matrix $A$ is defined as the determinant of the submatrix that results from the $i$-th row and $j$-th column from $A$.

$$M_{ij}=\det A_{ij}$$

Example

Consider a matrix $A$:

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 4& 5\\ 1& 0& 6\end{array}\right]$$

To calculate the minor $M_{12}$ (i.e., the minor of the element in the first row and the second column):

Remove the first row and the second column:

The submatrix is:

$$\left[\begin{array}{cc}0& 5\\ 1& 6\end{array}\right]$$

Compute the determinant of the submatrix:

$$M_{12}=\det \left[\begin{array}{cc}0& 5\\ 1& 6\end{array}\right]=0\cdot 6-5\cdot 1=-5$$