Page 135 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Theorem 1.7.1c: A triangular matrix is invertible if and only if its diagonal entries are all nonzero.

Theorem 1.7.1d: The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible
upper triangular matrix is upper triangular.
We will now prove these results using the adjoint formula for the inverse. We need a preliminary result.

THEOREM 2.1.3

If is an triangular matrix (upper triangular, lower triangular, or diagonal), then is the product of the

entries on the main diagonal of the matrix; that is, .

For simplicity of notation, we will prove the result for a lower triangular matrix

The argument in the case is similar, as is the case of upper triangular matrices.

Proof of Theorem 2.1.3 ( lower triangular case) By Theorem 2.1.1, the determinant of may be found by cofactor
expansion along the first row:

Once again, it's easy to expand along the first row:

where we have used the convention that the determinant of a matrix is .

EXAMPLE 8 Determinant of an Upper Triangular Matrix