# 204 - Theorem 2.5 Let A be an nn matrix. Then A is...

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1 Example: has the reduced row echelon form I n invertible. has the reduced row echelon form not invertible. Theorem 2.5 Let A be an n × n matrix. Then A is invertible if and only if the reduced row echelon form of A is I n . Proof ” For v R n , A v = 0 A 1 A v = A 1 0 = 0 v = 0 rank A = n by Theorem 1.8 reduced row echelon form of A is I n since A is n × n ” By Theorem 2.3 an invertible n × n matrix P s.t. PA = I n A = I n A = ( P 1 P ) A = P 1 ( PA ) = P 1 I n = P 1 A is invertible. Proof Example: augmented

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2 Proof By Theorem 2.5 (a) (b) , by Theorem 1.6 (b) (c) (d) (e) , and by Theorem 1.8 (b) (f) (g) (h) . “(a) (k)” (a) (b) By Theorem 2.3 elementary matrices E k , E k 1 , , E 1 such that E k E k 1 " E 1 A = I n A = E 1 " E k 1 E k = product of elementary matrices. (Invertible Matrix Theorem)
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## This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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204 - Theorem 2.5 Let A be an nn matrix. Then A is...

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