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Lecture 4

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University of Toronto Scarborough
Kathleen Smith

De▯nition: Any matrix that can be obtained from an identity ma- trix by means of one elementary row operation is called an elemen- tary matrix. Theorem: Let A 2 M n;kR) and let E 2 M n;n(R) be elementary. Multiplication of A on the left by E e▯ects the same elementary row operation on A that was performed on the identity matrix to obtain E. Note: If A s B, there are elementary matrices E ; E ;1▯▯▯2;E ‘ such that B = (E ▯▯‘E E )2. 1 De▯nition: An n▯n matrix A is invertible if there exists an n▯n matrix B such that B A = AB = I. The matrix B is called the inverse of A and is denoted by A . If A is not invertible, it is said to be singular. Note: We often write M nR) for M n;nR) when we are dealing with square matrices. Theorem: Let A 2 M (R).nIf B and C are matrices such that AB = C A = I, then B = C. In particular, if AB = B A = I, then B is the unique matrix with this property. Theorem: If A; B 2 M (R)nare invertible, then AB is invertible ▯1 ▯1 ▯1 and (AB) = B A . Fact: Every elementary matrix is invertible. Lemma: Let A 2 M (n). The linear sy
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