MAT1341-L16.Determinantes-VC

MAT1341-L16.Determinantes-VC - DETERMINANTS JOS E MALAG...

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Unformatted text preview: DETERMINANTS JOS E MALAG ON-L OPEZ In this lecture we will see how we can determine, without using a system of equations, if a collection of n vectors in R n is linearly independent. The idea is to associate to any square matrix a number, the determi- nant, which will contain such information. Definition of Determinant Given a square matrix A = ( a ij ) of order n , the submatrix A ij is obtained by deleting the i-th row and the j-th column of A . a 11 a 1 j 1 a 1 j a 1 j +1 a 1 n . . . . . . . . . . . . a i 11 a i 1 j 1 a i 1 j a i 1 j +1 a i 1 n a i 1 a i j 1 a ij a i j +1 a in a i +1 a i +1 j 1 a i +1 j a i +1 j +1 a i +1 n . . . . . . . . . . . . a m 1 a mj 1 a mj a mj +1 a mn Remark 1 . If A is an n n matrix, then any submatrix A ij is an ( n 1) ( n 1) matrix. Example 2. A = 2 3 10 5 1 7 9 4 Then A 21 = parenleftbigg 3 10 9 4 parenrightbigg A 12 = parenleftbigg 1 7 4 parenrightbigg A 31 = parenleftbigg 3 10 5 1 parenrightbigg A 13 = parenleftbigg 0 5 7 9 parenrightbigg 1 Definition 3. The determinant of a square n n matrix A , denoted as det( A ) or | A | , is defined recursively as follows: (1) If n = 1, i.e., A = ( a 11 ), then det( a 11 ) = a 11 (2) If n = 2, then det parenleftbigg a 11 a 12 a 21 a 22 parenrightbigg = a 11 a 22 a 12 a 21 (3) If n = 3, then det a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 det( A 11 ) a 12 det( A 12 ) + a 13 det( A 13 ) (4) If n > 3, then det a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . ....
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MAT1341-L16.Determinantes-VC - DETERMINANTS JOS E MALAG...

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