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MATH125 (99)
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Lecture 27

# MATH125 Lecture 27: 27

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School
University of Alberta
Department
Mathematics
Course
MATH125
Professor
Nikita
Semester
Winter

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ De▯nition. The determinant of an n▯n matrix A ▯ ▯a ▯ isijhe sum of n terms of the form ▯a ▯▯▯A , with plus and minus alternating 1j 1j where the entries 11; a12:::;a1n are from the ▯rst row of A. Thus 1+n ▯▯▯A ▯ a 11▯A 11▯ a12▯▯A 12 ▯ ▯▯▯ ▯ ▯▯▯▯ a1n▯▯▯A 1n Example. ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯ A ▯ ▯ ▯ ▯ ▯▯ ▯ : ▯ ▯▯ ▯ Solution. ▯▯ ▯▯▯▯ [ ] [ ] [ ] ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯▯▯A ▯ ▯ ▯ ▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯ ▯ ▯ ▯▯▯ ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯▯ ▯ ▯▯ ▯ ▯▯: Notation. ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ [ ] a b ▯▯▯ c d ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯ ▯ ▯a b ▯ ▯ ▯: ▯ c d ▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯ De▯nition. The ▯i;j▯-cofactor of A is the scalar C ▯ ▯▯▯▯ i+j▯▯▯A ; ij ij so that ▯▯▯A ▯ a 11 11 ▯ a12 12▯ ▯▯▯ ▯ a1n :1n ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ A▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ The Laplace Expansion Theorem. The determinant of an n ▯ n matrix A can be computed by a cofactor expansion across any row or down any column, so that we have ▯▯▯A ▯ a i1▯i1 C ▯i2▯i2▯ a C in in (cofactor expansion along the ith row) and ▯▯▯A ▯ a C1j 1j▯ a2j 2j▯ ▯▯▯ ▯ njC nj (cofactor expansion along the jth column). Example. ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯A ▯▯▯▯▯ ▯ ▯ ▯ ▯ ▯ A ▯ ▯ ▯ ▯ ▯▯ ▯ : ▯ ▯▯ ▯ Solution. ▯▯ ▯▯▯▯ ▯▯▯A ▯ a C 31 31▯ a32 32▯ a33 33 ▯ 3+1 3+2 3+3 ▯▯▯▯ a31▯▯A 31▯ ▯▯▯▯ a32▯▯A 32▯ ▯▯▯▯ a33▯▯A 33 ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯▯ ▯▯▯▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯▯▯▯ ▯ ▯ ▯ ▯▯: ▯ ▯▯ ▯ ▯▯ ▯ ▯ Example. ▯▯▯▯▯▯▯ ▯▯▯A ▯▯▯▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ A ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ : ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯▯ ▯ ▯ ▯ ▯ ▯▯ ▯ Solution. ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯
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