CorollaryComputing determinants by row reduction

Suppose the \((n,n)\)-matrix \(\Mtrx{A}\) can be row reduced to an upper triangular matrix \(\Mtrx{U}\) with diagonal entries \(d_1,\dots ,d_n\). If this row reduction used

\(r\) interchanges of rows; and
the multiplication of rows by (nonzero) numbers \(c_1,\dots ,c_k\), then

\[\DtrmnntOf{\Mtrx{A} } = (-1)^r\cdot \dfrac{d_1\cdots d_n}{c_1\cdots c_k}\]

Proof

We know that the determinant operation on \((n,n)\)-matrices is alternating and multilinear in the columns of matrices. Now we have \(\DtrmnntOf{\Mtrx{A} } =\DtrmnntOf{\MtrxTrnsps{A} }\) for every matrix \(\Mtrx{A}\). Therefore the determinant operation is alternating and multilinear in the rows of matrices as well.

Now if \(\Mtrx{A}\) is given, we know that we can row reduce it to upper triangular matrix \(\Mtrx{U}\) using suitable combinations of

an interchange of columns: the determinant switches its sign;
multiplying a column by a nonzero constant \(c\): the determinant gets multiplied by \(c\);
adding a multiple of one row to another: the determinant remains unchanged.

This implies the claim.