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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 have

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.

For a -matrix , we have

This is called the rule of Sarrus.

For an upper triangular matrix

we have
In particular, for the

identity matrix we get .

This follows with a simple induction directly from the recursive definition of the determinant.