Let be a
field,
and let denote an
-matrix
over with entries . For
,
let denote the -matrix, which arises from , when we remove the first column and the -th row. Then one defines recursively the determinant of by
The determinant is only defined for square matrices. For small , the determinant can be computed easily.
For a -matrix, we can use the rule of Sarrus to compute the determinant. We repeat the first column as the fourth column and the second column as the fifth column. The products of diagonals from up to down enter with a positive sign, and the products of the other diagonals enter with a negative sign.