【文章內(nèi)容簡介】 or a matrix with zero rows 2. It is easy to calculate the determinant of an upper triangular matrix (by Theorem ) or a matrix with zero rows (det = 0) ? Ex 2: Evaluation a determinant using elementary row operations ????????????3102211032A ?)de t( ?? ASol: 1 , 22 3 10 1 2 2de t( ) 1 2 2 2 3 100 1 3 0 1 3IA??? ? ??? ? ???A1A11d e t( ) d e t( ) d e t( ) d e t( )AAAA??? ? ?? Notes: ( 1 )2 , 31 2 27 0 1 2 7 ( 1 ) 70 0 1A ?????? ? ? ? ? ??1()( 2 ) 71 , 2 21 2 2 1 2 210 7 1 4 ( 1 ) ( ) 0 1 2( 1 / 7 )0 1 3 0 1 3A M ???????? ? ? ???? ? ??? Notes: 213 2 2 343de t ( ) de t ( )11de t ( ) de t ( ) de t ( ) de t ( )7 ( 1 / 7 )de t ( ) de t ( )AAA A A AAA?? ? ? ???2A 3A4A Cofactor Expansion Row Reduction Order n Additions Multiplications Additions Multiplications 3 5 9 5 10 5 119 205 30 45 10 3,628,799 6,235,300 285 339 ? Comparison between the number of required operations for the two kinds of methods to calculate the determinant ※ When evaluating a determinant by hand, you can sometimes save steps by integrating this two kinds of methods (see Examples 5 and 6 in the next three slides) ? Ex 5: Evaluating a determinant using column reduction and cofactor expansion ???????????????603142253A Sol: ( 2 )1 , 3313 5 2 3 5 4de t( ) 2 4 1 2 4 33 0 6 3 0 054( 3 ) ( 1 ) ( 3 ) ( 1 ) ( 1 ) 343ACA?? ? ?? ? ? ? ????? ? ? ? ? ? ??※ is the counterpart column operation to the row operation (),kijAC (),kijA ? Ex 6: Evaluating a determinant using both row and column reductions and cofactor expansion Sol: ???????????????????0231134213321011231223102A( 1 ) ( 1 )2 , 4 2 , 5 222 0 1 3 2 2 0 1 3 22 1 3 2 1 2 1 3 2 1de t ( ) 1 0 1 2 3 1 0 1 2 33 1 2 4 3 1 0 5 6 41 1 3 2 0 3 0 0 0 12 1 3 21 1 2 3( 1)( 1)1 5 6 43 0 0 1AAA????? ? ? ??? ??? ? ?????? ( 3 ) ( 1 )4 , 1 2 , 144138 1 3 28 1 3 0 0 58 1 2 3( 1) ( 1 ) 8 1 2 = 8 1 2 13 5 6 413 5 6 13 5 60 0 0 181 5( 1)13 5 ( 5 ) ( 27 ) 135A C A??????? ? ? ? ? ? ?????????? ? Theorem : Conditions that yield a zero determinant (a) An entire row (or an entire column) consists of zeros (b) Two rows (or two columns) are equal (c) One row (or column) is a multiple of another row (or column) If A is a square matrix and any of the following conditions is true, then det(A) = 0 ? Notes: For conditions (b) or (c), you can also use elementary row or column operations to create an entire row or column of zeros and obtain the results by Theorem ※ Thus, we can conclude that a square matrix has a determinant of zero if and only if it is row (or column) equivalent to a matrix that has at least one row (or column) consisting entirely of zeros (For (b) and (c), based on the mathematical induction (數(shù)學(xué)歸納法 ), perform the cofactor expansion along any row or column other than these two rows or columns) (Perform the cofactor expansion along the zero row or column) 0654000321? 0063052041? 0654222111?0261251241? 0642654321????061235102481?? Ex: Properties of Determinants ? Notes: )de t ()de t ()de t ( BABA ???? Theorem : Determinant of a matrix product (1) (3) (4) 11 12 13 11 12 1321 22 23 21 22 2331 32 33 31 32 33a a a a a aa a a b b ba a a a a a??11 12 1321 21 22 22 23 2331 32 33a a aa b a b a ba a a? ? ? det(AB) = det(A) det(B) (There is an example to verify this property on Slide ) (Note that this property is also valid for all rows or columns other than the second row) , , , ,( ) ( ) ( ) ( )( ) ( ) ( ) ( ), , , ,de t ( ) de t ( ) de t ( ) a nd d e t ( ) de t ( ) de t ( ) 1 de t ( ) de t ( ) de t ( ) a nd d e t ( ) de t ( ) de t ( )de t ( ) de t ( ) de t ( ) a nd d e t ( ) de t ( ) de t ( ) 1i j i j i j i jk k k ki i i ik k k ki j i j i j i jE A E A E A A EE A E A E A k A E kE A E A E A A E? ? ? ? ? ?? ? ? ?? ? ? ? ?????(2) 1 2 1 2de t( ) de t( ) de t( ) de t( )nnA A A A A A?(Verified by Ex 1 on the next slide) ? Ex 1: The determinant of a matrix product ?????????? ??101230221A??????????????213210102B7101230221|| ????A11213210102|| ?????BSol: Find |A|, |B|, and |AB| ??