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Unformatted text preview: Lecture 22 Andrei Antonenko March 31, 2003 1 Properties of determinants This lecture we will start studying a properties of determinants, and algorithms of computing them. Lets recall, that we defined a determinant by the following way: det a 11 a 12 ... a 1 n a 21 a 22 ... a 2 n . . . . . . . . . . . . . . . . . . a n 1 a n 2 ... a nn = X all permutations of n elements sgn( ) a 1 (1) a 2 (2) a n ( n ) . (1) Now well start with properties of determinants. Theorem 1.1 (1st elementary row operation). If 2 rows of a matrix A are interchanged, then the determinant changes its sign. Proof. Suppose B arises from A by interchanging rows r and s of A , and suppose r < s . Then we have that b rj = a sj and b sj = a rj for any j , and a ij = b ij if i 6 = r,s . Now det B = X all permutations of n elements sgn( ) b 1 (1) b r ( r ) ...b s ( s ) ...b n ( n ) = X all permutations of n elements sgn( ) a 1 (1) a s ( r ) ...a r ( s ) ...a n ( n ) = X all permutations of n elements sgn( ) a 1 (1) a r ( s ) ...a s ( r ) ...a n ( n ) . The permutation ( (1) ... ( s ) ... ( r ) ... ( n )) is obtained from ( (1) ... ( r ) ... ( s ) ... ( n )) by interchanging 2 numbers, so its sign is different, and det B =- det A . Theorem 1.2 (Determinant of a matrix with 2 equal rows). If 2 rows of a matrix are equal, then its determinant is equal to 0. Proof. Suppose rows r and s of matrix A are equal. Interchange them to obtain matrix B ....
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
- Spring '03