Here 2 y 0 so y sin 1 and since x y 3 x cos

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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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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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