Let be a square matrix over a field .
- Compute the determinant of each type of elementary matrix.
- Prove that is nonzero if and only if is row-equivalent to the identity matrix. Suppose there is a sequence of row operations containing row interchanges and row multiplications, using the nonzero constants . Prove that .
We discussed elementary matrices in this previous exercise.
Note that if is not the identity, then there must exist an element such that . (Otherwise, we can show by induction that .) Now recalling the naive formula for computing , note that if is an upper- or lower-triangular matrix, then is merely the product of the diagonal entries.
In particular, we have , and if . Note also that the row operation of interchanging two rows is equivalent to three operations which add a multiple of one row to another and one scalar row multiplication by -1. Since determinants are multiplicative, the determinant of the elementary matrix achieving this operation is -1.
Suppose . In this previous exercise, we saw that the columns of are linearly independent. In particular, all columns in the reduced row echelon form of are pivotal, and so is row equivalent to the identity matrix. Conversely, suppose is row equivalent to the identity matrix. Then the columns of are linearly independent, and so .
Suppose now that , where is a product of elementary matrices. Since determinants are multiplicative, it is clear that has the given form.