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MATH125 (99)
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Lecture 20

# MATH125 Lecture 20: 20

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

Description
▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯ Elementary matrices De▯nition. An elementary matrix is one that is obtained by perform- ing a single elementary row operation on an identity matrix. ▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ Example. ▯▯▯ ▯▯▯▯▯▯ 2 3 ▯ ▯ ▯ 4 5 E 1 ▯ ▯ ▯ ▯▯ ▯ ▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ 3▯▯▯▯▯ I ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯ ▯▯ ▯▯▯ ▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯▯1▯▯▯ E ▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ 2 3 ▯ ▯ ▯ E ▯ 4 ▯ ▯ ▯ 5 ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯; ▯ 2 ▯ ▯ ▯ ▯▯▯ 2 3 ▯ ▯ ▯ E ▯ 4 ▯ ▯ ▯ 5 ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯ ▯ : 3 ▯ ▯ ▯ Question. ▯▯▯ 2 3 a b c A ▯ 4 d e f 5: g h i ▯▯▯▯▯▯▯ E 1; E A2 E A3▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯ A▯ Solution. ▯▯ ▯▯▯▯ 2 3 a b c 4 5 E1A ▯ d e f ; g ▯ ▯a h ▯ ▯b i ▯ ▯c 2 3 d e f 4 5 E2A ▯ a b c ; g h i 2 3 a b c 4 5 E3A ▯ d e f : ▯g ▯h ▯i ▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯ ▯ ▯▯ ▯▯▯ ▯ ▯▯▯▯▯▯▯▯ E1A▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯ ▯▯▯ ▯ ▯▯▯▯▯▯2▯ E A▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯ ▯▯ ▯ ▯▯▯▯▯▯▯▯3E A▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ Theorem. If an elementary row operation is performed on an m▯n matrix A, the resulting matrix can be written as EA where E is the matrix of size m▯m created by performing the same row operation on the identity matrmx I . Main observation: ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯ E ▯▯ ▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯mI▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯ E ▯▯▯▯ ▯▯mI ▯ ▯▯▯▯▯ ▯▯▯▯▯ ▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ F ▯▯▯▯ ▯▯▯▯ FE ▯ I ▯ ▯▯▯▯▯ E ▯▯▯ F ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ m ▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯ EF ▯mI ▯ ▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ E ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯ Theorem. A square matrix A of size n▯n is invertible if and only if A is row equivalent to the identity natrix I and in this case any sequence of elementary row operations that reduces A to I also transforms I ▯1 n n to A . Proof. ▯▯▯▯▯▯▯ ▯▯▯▯ A ▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯ Ax ▯ ▯ ▯▯▯ ▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯▯▯ ▯▯▯▯ ▯▯▯▯▯ ▯▯▯ ▯▯ ▯▯▯▯ ▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ ▯▯▯▯▯▯▯
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