DLA Theory

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Section 8.5 Theory

In this section.

Subsection 8.5.1 Basic properties of determinants

Subsection 8.5.1 Basic properties of determinants

The following justifies our definition of the determinant as the common value of all cofactor expansions of a matrix.

Theorem 8.5.1. Uniformity of cofactor expansions.

Every cofactor expansion of a given square matrix, whether along a row or a column, evaluates to the same value.

Proof.

The proof of this theorem is beyond the scope of this course; instead see [2, 3].

Finally, let's record the determinants of special forms of matrices we discussed in Subsection 8.3.5. However, we omit the proofs since we have already considered in detail the patterns behind the proofs in that earlier discussion.

Proposition 8.5.2. Determinants of special forms.

For a matrix that is diagonal or triangular, the determinant is equal to the product of the diagonal entries.
For a scalar matrix, $\det (k I) = k^n\text{.}$
$\det \zerovec = 0$ for a square zero matrix.
$\det I = 1\text{.}$