The Gauss-Jordan elimination process is also one of the best methods for calculating determinants. For convenience, we summarize the effects of the elementary row operations [§4.18] on the value of the determinant.
Operation |
Effect |
Proof |
Interchange 2 rows |
detA ® -detA |
Theorem 5.3 |
Multiply a row by scalar t |
detA ® t detA |
Axiom 1 |
Add to one row a scalar multiple of another |
detA ® detA |
Theorem 5.3 |
We shall show that the last operation alone is sufficient to transform an upper triangular matrix into a diagonal one.
An upper triangular matrix is any n ´ n matrix of the form
so that all elements below the diagonal
vanish, i.e.,
For an n ´ n upper diagonal matrix U,
It is easy to see that if we apply the backward substitution process of the Gauss-Jordan method [see §4.18] to (5.6), we can transform U to the equivalent form
provided