Final - 4 Determinants Lecture 12 Let T : R 2 → R 2 be a...

Info iconThis preview shows pages 1–3. 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 DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4 Determinants Lecture 12 Let T : R 2 → R 2 be a linear transformation with the standard matrix A = • a 11 a 12 a 21 a 22 ‚ . Recall that the determinant of a 2 × 2 matrix h a 11 a 12 a 21 a 22 i is defined by det • a 11 a 12 a 21 a 22 ‚ = fl fl fl fl a 11 a 12 a 21 a 22 fl fl fl fl = a 11 a 22- a 12 a 21 . It is easy to see that for vectors u , v , w ∈ R 2 , we have det[ a u + b v , w ] = a det[ u , w ] + b det[ v , w ] , det[ u ,a v + b w ] = a det[ u , v ] + b det[ u , w ] . This means that det is linear in each column vector variables. Moreover, switching two columns (rows) changes the sign, i.e., det[ u , v ] =- det[ v , u ] . Let A = [ a ij ] be an n × n square matrix. For a fixed ( i,j ), where 1 ≤ i ≤ m and 1 ≤ j ≤ n , let A ij = ( n- 1) × ( n- 1) submatrix of A obtained by deleting the i th row and j th column of A. Definition 4.1. Let A = [ a ij ] be an n × n square matrix. The determinant of A is a number det A , inductively defined by det A = a 11 det A 11- a 12 det A 12 + ··· + (- 1) n +1 a 1 n det A 1 n = a 11 det A 11- a 21 det A 21 + ··· + (- 1) n +1 a n 1 det A n 1 . The motivation of this definition is as follows: Write A = [ a 1 , a 2 , ··· , a n ], Note that a 1 = a 11 + a 21 + ··· + a n 1 , where a 11 =      a 11 . . .      , a 21 =      a 21 . . .      , a n 1 =      . . . a n 1      . Since det A is linear in each column vectors of the matrix A , i.e., det[ a u + b v , a 2 ,..., a n ] = a det[ u , a 2 ,..., a n ] + b det[ v , a 2 ,..., a n ], we have det A = det[ a 11 + a 21 + ··· + a n 1 , a 2 ,..., a n ] = det[ a 11 , a 2 ,..., a n ] + det[ a 21 , a 2 ,..., a n ] + ··· + det[ a n 1 , a 2 ,..., a n ] = det • a 11 * A 11 ‚- det • a 21 * A 21 ‚ + ··· + (- 1) n- 1 det • a n 1 * A n 1 ‚ = a 11 det A 11- a 21 det A 21 + ··· + (- 1) n +1 a n 1 det A n 1 . 15 Theorem 4.2. Let A = [ a ij ] be an n × n matrix. The ( i,j )-cofactor of A (1 ≤ i,j ≤ n ) is the number C ij = (- 1) i + j det A ij . Then det A = a 11 C 11 + a 12 C 12 + ··· + a 1 n C 1 n = a 11 C 11 + a 21 C 21 + ··· + a n 1 C n 1 . Example 4.1. For any 3 × 3 matrix, fl fl fl fl fl fl a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 fl fl fl fl fl fl = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32- a 11 a 23 a 32- a 12 a 21 a 33- a 13 a 22 a 31 . Theorem 4.3 (Cofactor Expansion Formula) . For any n × n square matrix A = [ a ij ] , det A = a i 1 C i 1 + a i 2 C i 2 + ··· + a in C in = a 1 j C 1 j + a 2 j C 2 j + ··· + a nj C nj . In other words, det A T = det A. Proposition 4.4. fl fl fl fl fl fl fl fl fl a 11 a 12 ··· a 1 n a 22 ··· a 2 n ....
View Full Document

This note was uploaded on 01/28/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

Page1 / 44

Final - 4 Determinants Lecture 12 Let T : R 2 → R 2 be a...

This preview shows document pages 1 - 3. Sign up to view the full document.

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