Unformatted text preview: lower triangular, so is A11 and, as such, the induction
hypothesis applies to A11 . In other words, det (A11 ) is the product of the entries along A11 ’s
main diagonal. Now, the entries on the main diagonal of A11 are the entries a22 , a33 , . . . ,
a(k+1)(k+1) from the main diagonal of A. Hence,
det(A) = a11 det (A11 ) = a11 a22 a33 · · · a(k+1)(k+1) = a11 a22 a33 · · · a(k+1)(k+1)
We have det(A) is the product of the entries along its main diagonal. This shows P (k + 1) is
true, and, hence, by induction, the result holds for all n × n upper triangular matrices. The
n × n identity matrix In is a lower triangular matrix whose main diagonal consists of all 1’s.
Hence, det (In ) = 1, as required. 9.4 The Binomial Theorem 9.4 581 The Binomial Theorem In this section, we aim to prove the celebrated Binomial Theorem. Simply stated, the Binomial
Theorem is a formula for the expansion of quantities (a + b)n for natural numbers n. In Elementary
and Intermediate Algebra, you should have seen speciﬁc...
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