Minor (mathematics)

In linear algebra, the minor of a matrix is the determinant of the matrix obtained after deleting a row <math>i</math> and column <math>j</math>, and is written <math>M_{ij}</math>.

For example, given the matrix

<math>\begin{pmatrix}

1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ \end{pmatrix}</math>

and suppose we wish to find the minor of the (2,3) entry. We consider the matrix with row 2 and column 3 removed (note the following is not standard notation!):

<math>\begin{pmatrix}

1 & 4 & ! \\ ! & ! & ! \\ -1 & 9 & ! \\ \end{pmatrix}</math>

This gives:

<math>M_{23}=\begin{vmatrix}

1 & 4 \\ -1 & 9 \\ \end{vmatrix}=9+4=13</math>

Specially signed minors, known as cofactors, are useful in calculating the inverse of small matrices.