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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. Let’s 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 σ sgn( σ ) a 1 σ (1) a 2 σ (2) ··· a nσ ( n ) . (1) Now we’ll 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 σ sgn( σ ) b 1 σ (1) ··· b rσ ( r ) ...b sσ ( s ) ...b nσ ( n ) = X all permutations of n elements σ sgn( σ ) a 1 σ (1) ··· a sσ ( r ) ...a rσ ( s ) ...a nσ ( n ) = X all permutations of n elements σ sgn( σ ) a 1 σ (1) ··· a rσ ( s ) ...a sσ ( r ) ...a nσ ( n ) . The permutation ( σ (1) ...σ ( s ) ...σ ( r ) ...σ ( n )) is obtained from ( σ (1) ...σ ( r ) ...σ ( s ) ...σ ( 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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- Spring '03
- ........., Invertible matrix, Ann, Det