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Unformatted text preview: row, then det(A ) = det(A)
• If A has two identical rows, or a row consisting of all 0’s, then det(A) = 0.
• If A is upper or lower triangular,a then det(A) is the product of the entries on the main
• If B is an n × n matrix, then det(AB ) = det(A) det(B ).
• det (An ) = det(A)n for all natural numbers n.
• A is invertible if and only if det(A) = 0. In this case, det A−1 =
det(A) See Exercise 5 in 8.3.
See page 483 in Section 8.3. Unfortunately, while we can easily demonstrate the results in Theorem 8.7, the proofs of most of
these properties are beyond the scope of this text. We could prove these properties for generic 2 × 2
or even 3 × 3 matrices by brute force computation, but this manner of proof belies the elegance and 8.5 Determinants and Cramer’s Rule 511 symmetry of the determinant. We will prove what few properties we can after we have developed
some more tools such as the Principle of Mathematical Induction in Section 9.3.2 For the moment,
let us demonstrate some of the properties listed in Theorem 8.7 on the ma...
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