{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

03-31-03 - Lecture 22 Andrei Antonenko 1 Properties of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 4

03-31-03 - Lecture 22 Andrei Antonenko 1 Properties of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online