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Unformatted text preview: Page 1 of 4 November 17 th , 2009 MAT240: Abstract Linear Algebra Lecture: Question 1: Can we choose P, Q so that PAQ is simpler? Question 2: Whats simpler? Answer to Question 2: Claim: Let ? = ? then ???? = ? , where k is the number of row/columns of I ? : ? ? , ? = = ( ) ? = ?? 1 ? ??? ? = ? Elementary Matrices Three kinds , 1 , , 2 , , , 3 1. , 1 = interchange row/column i with row/column j , 1 = with rows i,j interchanged. , 1 = with columns i,j interchanged. Claim: , 1 is invertible and , 1 1 = , 1 Proof: , 1 , 1 = 2. , 2 = multiply row or column I by a constant c (c cannot equal 0). If c = 0, then the matrix is not invertible....
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