【正文】 Chapter 3 Determinants The Determinant of a Matrix Evaluation of a Determinant using Elementary Row Operations Properties of Determinants Application of Determinants: Cramer’s Rule ※ The determinant is NOT a matrix operation ※ The determinant is a kind of information extracted from a square matrix to reflect some characteristics of that square matrix ※ For example, this chapter will discuss that matrices with a zero determinant are with very different characteristics from those with nonzero determinants ※ The motives to calculate determinants are to identify the characteristics of matrices and thus facilitate the parison between matrices since it is impossible to investigate or pare matrices entry by entry ※ The similar idea is to pare groups of numbers through the calculation of averages and standard deviations ※ Not only the determinant but also the eigenvalues and eigenvectors are the information that can be used to identify the characteristics of square matrices The Determinant of a Matrix ? The determinant (行列式 ) of a 2 2 matrix: ? Note: 1. For every SQUARE matrix, there is a real number associated with this matrix and called its determinant 2. It is mon practice to omit the matrix brackets ???????22211211aaaaA12212211||)d e t ( aaaaAA ???????????22211211aaaa22211211aaaa ? Historically speaking, the use of determinants arose from the recognition of special patterns that occur in the solutions of linear systems: ? Note: 1. x1 and x2 have the same denominator, and this quantity is called the determinant of the coefficient matrix A 2. There is a unique solution if a11a22 – a21a12 = |A| ≠ 0 11 1 12 2 121 1 22 2 21 22 2 12 2 11 1 211211 22 21 12 11 22 21 12 a nd a x a x ba x a x bb a b a b a b axxa a a a a a a a?????????? ? ? ? Ex. 1: The determinant of a matrix of order 2 734)3(1)2(221 32 ??????? 044)1(4)2(22412 ?????330)2/3(2)4(042 2/30 ??????? Note: The determinant of a matrix can be positive, zero, or negative ? Minor (子行列式 ) of the entry aij: the determinant of the matrix obtained by deleting the ith row and jth column of A ? Cofactor (餘因子 ) of aij: ijjiij MC ??? )1(nnjnjnnnijijiinijijiinjjijaaaaaaaaaaaaaaaaaM???????????????)1()1(1)1()1)(1()1)(1(1)1()1()1)(1()1)(1(1)1(1)1(1)1(11211?????????????????※ Mij is a real number ※ Cij is also a real number ? Ex: ???????????333231232221131211aaaaaaaaaA3332131221 aaaaM ??21211221 )1( MMC ????? ?3331131122 aaaaM ?22222222 )1( MMC ??? ?? Notes: Sign pattern for cofactors. Odd positions (where i+j is odd) have negative signs, and even positions (where i+j is even) have positive signs. (Positive and negative signs appear alternately at neighboring positions.) ????????????????????????????????????????????????????? ? Theorem : Expansion by cofactors (餘因子展開 ) 1 1 2 21( a ) d e t( ) | |nij ij i i i i in injA A a C a C a C a C?? ? ? ? ? ??(cofactor expansion along the ith row, i=1, 2,…, n) 1 1 2 21( b ) d e t( ) | |ni j i j j j j j n j n jiA A a C a C a C a C?? ? ? ? ? ??(cofactor expansion along the jth column, j=1, 2,…, n) Let A be a square matrix of order n, then the determinant of A is given by or ※ The determinant can be derived by performing the cofactor expansion along any row or column of the examined matrix ? Ex: The determinant of a square matrix of order 3 ???????????333231232221131211aaaaaaaaaA11 11 12 12 13 1321 21 22 22 23 2331 31 32 32 33 3311 11 21 21 31 3112 12 22 22 3de t ( ) ( f i r st r ow e xpa nsi on) ( se c ond r ow e xpa nsi on) ( t hi r d r ow e xpa nsi on) ( f i r st c ol um n e xpa nsi on)A a C a C a Ca C a C a Ca C a C a Ca C a C a Ca C a C a? ? ? ?? ? ?? ? ?? ? ?? ? ?2 3213 13 23 23 33 33 ( se c ond c ol um n e xpa nsi on) ( t hi r d c ol um n e xpa nsi on)Ca C a C a C? ? ? ? Ex 3: The determinant of a square matrix of order 3 ?)det ( ?? A0 2 13 1 2401A????????????110 21)1( 1111 ????? ?CSol: 5)5)(1(14 23)1( 2112 ?????? ?C404 13)1( 3113 ???? ?C14)4)(1()5)(2()1)(0()d e t ( 131312121111????????? CaCaCaA ? Alternative way to calculate the determinant of a square matrix of order 3: 11 12 1321 22 2331 32 33a a aA a a aa a a????????? 32313332312221232221