Chapter 3 Determinants 3.1The Determinant of a Matrix3.2Evaluation of a Determinant using Elementary Row Operations 3.3Properties of Determinants3.4Introduction to Eigenvalues 3.5Application of Determinants3.1 3.2 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 In this chapter, for example, we will discuss that matrices with zero determinant are with very different characteristics from those with non-zero determinant The motive to find this information is to identify the characteristics of matrices and thus facilitate the comparison between matrices since it is impossible to compare matrices entry by entry 3.3 3.1 The Determinant of a Matrix The determinant 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 common practice to delete the matrix brackets ((

=22 2112 11a aa aA12 21 22 11| | ) det( a a a a A A = = =((

22 2112 11a aa a22 2112 11a aa a3.4 Ex. : The determinant of a matrix of order 2 2 13 2 2 41 24 23 0 Note: The determinant of a matrix can be positive, zero, or negative ) 3 ( 1 ) 2 ( 2 = 3 4 + = 7 =) 1 ( 4 ) 2 ( 2 = 4 4 = 0 =) 3 ( 2 ) 4 ( 0 = 6 0 = 6 =3.5 Minor of the entry aij: the determinant of the matrix obtained by deleting the i-th row and j-th column of A Cofactor of aij: ijj iijM C+ = ) 1 (nn j n j n nn i j i j i in i j i j i in j jija a a aa a a aa a a aa a a a aM ) 1 ( ) 1 ( 1) 1 ( ) 1 )( 1 ( ) 1 )( 1 ( 1 ) 1 () 1 ( ) 1 )( 1 ( ) 1 )( 1 ( 1 ) 1 (1 ) 1 ( 1 ) 1 ( 1 12 11+ + + + + + + + = Mij is a real number Cij is a real number 3.6 Ex: ((((

=33 32 3123 22 2113 12 11a a aa a aa a aA33 3213 1221a aa aM= 21 211 221) 1 ( M M C = = +33 3113 1122a aa aM =22 222 222) 1 ( M M C = =+ 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.) ((((((((

+ + + + + + + + + + + + + 3.7 Theorem 3.1: Expansion by cofactors 1 1 2 21(a) det( ) | |nij ij i i i i in injA A a C a C aC aC== = = + + +(cofactor expansion along the i-th row, i=1, 2,, n) 1 1 2 21(b) det( ) | |nij ij j j j j nj njiA A a C aC a C aC== = = + + +(cofactor expansion along the j-th 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 3.8 Ex: The determinant of a matrix of order 3 ((((

=33 32 3123 22 2113 12 11a a aa a aa a aA11 11 12 12 13 1321 21 22 22 23 2331 31 32 32 33 3311 11 21 21 31 3112 12 22 22 3det( )(first row expansion) (second row expansion) (third row expansion) (first column expansion)A aC aC aCaC a C a CaC a C aCaC aC aCaC a C a = + += + += + += + += + +2 3213 13 23 23 33 33 (second column expansion) (third column expansion)CaC a C aC = + +3.9 Ex :The determinant of a matrix of order 3 ? ) det( = A0 2 13 1 24 0 1A ( (= ( ( 11 02 1) 1 (1 111 = =+CSol: 5 ) 5 )( 1 (1 42 3) 1 (2 112= = =+C40 41 3) 1 (3 113= =+C14) 4 )( 1 ( ) 5 )( 2 ( ) 1 )( 0 () det(13 13 12 12 11 11=+ + =+ + = C a C a C a A3.10 Alternative way to calculate the determinant of a matrix of order 3:11 12 1321 22 2331 32 33a a aA a a aa a a ( (=( ( 32 31 33 32 3122 21 23 22 2112 11 13 12 11a a a a aa a a a aa a a a a12 21 33 11 23 3213 22 31 32 21 13 31 23 12 33 22 11 | | ) det(a a a a a aa a a a a a a a a a a a A A + + = = Add these three products Subtract these three products 3.11 Ex: Recalculate the determinant of the matrix A in Ex 3 0 2 13 1 24 0 1A ( (= ( ( 0 23 14 0 4 0 det( ) | | 0 16 0 ( 4) 0 6 14 A A = =+ + =16 0 06 This method is only valid for matrices with the order of 3 3.12 Ex :The determinant of a matrix of order 4 (((((

=2 0 4 33 0 2 02 0 1 10 3 2 1A? ) det( = A3.13 Sol: ) )( 0 ( ) )( 0 ( ) )( 0 ( ) )( 3 ( ) det(43 33 23 13C C C C A + + + =2 4 33 2 02 1 1) 1 ( 33 1 =+133C =| |39) 13 )( 3 () 7 )( 1 )( 3 ( ) 4 )( 1 )( 2 ( 0 34 31 1) 1 )( 3 (2 32 1) 1 )( 2 (2 42 1) 1 )( 0 ( 33 2 2 2 1 2== + + =((

+ + =+ + + By comparing Ex 4 with Ex 3, it is apparent that the computational effort for the determinant of 44 matrices is much higher than that of 33 matrices. In the next section, we will learn a more efficient way to calculate the determinant. 3.14 Upper triangular matrix: Lower triangular matrix: Diagonal matrix: All entries below the main diagonal are zeros All entries above the main diagonal are zeros All entries above and below the main diagonal are zeros (((

3323 2213 12 110 00aa aa a a(((

33 32 3122 211100 0a a aa aaEx: upper triangular (((

3322110 00 00 0aaalower triangulardiagonal 3.15 Theorem 3.2: (Determinant of a Triangular Matrix) IfA is an n n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is nna a a a A A 33 22 11| | ) det( = = On the next slide, we only take the case of upper triangular matrices for example to prove Theorem 3.2. It is straightforward to apply the following proof for the cases of lower triangular and diagonal matrices. 3.16 Ex : Find the determinants of the following triangular matrices (a)(((((

=3 3 5 10 1 6 50 0 2 40 0 0 2A(b)(((((((

=2 0 0 0 00 4 0 0 00 0 2 0 00 0 0 3 00 0 0 0 1B|A| = (2)(2)(1)(3) = 12|B| = (1)(3)(2)(4)(2) = 48(a) (b) Sol: 3.17 3.2 Evaluation of a determinant using elementary row operations Theorem 3.3: Elementary row operations and determinants ,(a) ( )det( ) det( )i jB I A B A = = Let A and B be square matrices ( )(b) ( ) det( ) det( )kiB M A B k A = =( ),(c) ( ) det( ) det( )ki jB A A B A = = Notes: The above three properties remains valid if elementary column operations are performed to derive column-equivalent matrices The computational effort to calculate the determinant of a square matrix with a large number of n is unacceptable. In this section, I will show how to reduce the computational effort by using elementary operations 3.18 1 2 30 1 4 det( ) 21 2 1A A ( (= = ( ( Ex: 114 8 120 1 4 det( ) 81 2 1A A ( (= = ( ( 220 1 41 2 3 det( ) 21 2 1A A ( (= = ( ( 331 2 32 3 2 det( ) 21 2 1A A ( (= = ( ( (4)111( )det( ) 4det( ) (4)( 2) 8A M AA A= = = = 2 1, 22( )det( ) det( ) ( 2) 2A I AA A= = = =( 2)31,23( )det( ) det( ) 2A A AA A= = = 3.19 Row reduction method to calculate the determinant 1. A row-echelon form of a square matrix is always upper triangular 2. It is easy to calculate the determinant of an upper triangular matrix Ex 2:Evaluation a determinant using elementary row operations (((

=3 1 02 2 110 3 2A ? ) det( = ASol: 1,22 3 10 1 2 2det( ) 1 2 2 2 3 100 1 3 0 1 3IA = A1A11det( ) det( ) det( ) det( )A AA A= = Notes: 3.20 ( 1)2,31 2 27 0 1 2 7( 1) 70 0 1A = = 1( )( 2)71,221 2 2 1 2 210 7 14 ( 1) ( )0 1 2(1/ 7)0 1 3 0 1 3AM Notes: 2 13 2 2 34 3det( ) det( )1 1det( ) det( )det( ) det( )7 (1/ 7)det( ) det( )A AA A A AA A== ==2A3A4A3.21 Cofactor ExpansionRow Reduction Order nAdditions Multiplications Additions Multiplications 359 5 105119205 30 45103,628,799 6,235,300 285 339 Comparison between the number of required operations for the two kinds of methods to calculate the determinant 3.22 3.3 Properties of Determinants Notes: ) det( ) det( ) det( B A B A + = + Theorem 3.5: Determinant of a matrix product det (AB) = det (A) det (B)1 2 1 2det( ) det( )det( ) det( )n nA A A A A A =3.23 Ex : The determinant of a matrix product ((((

=1 0 12 3 02 2 1A((((

=2 1 32 1 01 0 2B71 0 12 3 02 2 1| | == A112 1 32 1 01 0 2| | = = BSol: Find|A|, |B|, and |AB|3.24 ((((

=((((

((((

=1 1 510 1 61 4 82 1 32 1 01 0 21 0 12 3 02 2 1AB8 4 1| | 6 1 10 775 1 1AB = = |AB| = |A| |B| Check: 3.25 Ex: ?1 2 2 1 2 2 1 2 20 3 2 1 1 2 1 2 01 0 1 1 0 1 1 0 1A B C = = + = +2 1 2 2 2 32 1 2 2 2 32 1 2 2 2 32 2 1 2 1 2| | 0( 1) 3( 1) 2( 1)0 1 1 1 1 02 2 1 2 1 2| | 1( 1) 1( 1) 2( 1)0 1 1 1 1 02 2 1 2 1 2| | 1( 1) 2( 1) 0( 1)0 1 1 1 1 0ABC+ + ++ + ++ + + = + + = + + = + + Pf: 3.26 Ex 2: 10 20 40 1 2 430 0 50 ,if3 0 5 5,find | |20 30 10 2 3 1A A ( (= = ( ( Sol: ((((

=1 3 25 0 34 2 110 A5000 ) 5 )( 1000 (1 3 25 0 34 2 1103= = = A Theorem 3.6: Determinant of a scalar multiple of a matrix If A is an n n matrix and c is a scalar, thendet (cA) = cn det (A) (can be proven by repeatedly use the fact that ) ( )if( ) | | | |kiB M A B k A = =3.27 Theorem 3.7: (Determinant of an invertible matrix) A square matrix A is invertible (nonsingular) if and only if det (A) = 0 3.28 Ex 3: Classifying square matrices as singular or nonsingular = 0 A((((

=1 2 31 2 31 2 0A((((

=1 2 31 2 31 2 0BA has no inverse (it is singular) = = 0 12 BB has inverse (it is nonsingular) Sol: 3.29 Ex 4: ?1=A((((

=0 1 22 1 03 0 1A? =TA (a) (b) 40 1 22 1 03 0 1| | = = A 41 11= = AA4 = =A ATSol: Theorem 3.8: Determinant of an inverse matrix 11If is invertiblethendet( )det( )A AA= , Theorem 3.9: Determinant of a transpose Ifis a square matrix, thendet( ) det( )TA A A =3.30 The similarity between the noninvertible matrix and the real number 0 Matrix AReal number c Invertible Noninvertible 1 1det( ) 01 exists and det( )det( )AA AA ==11det( ) 0 does not exist1 1det( )det( ) 0AAAA=| |= = |\ .1 101 exists and=cc cc =110 does not exist1 1= =0cccc=| | |\ .3.31 Ex 5: Which of the following system has a unique solution? (a) 4 2 34 2 31 23 2 13 2 13 2 = += + = x x xx x xx x

(b) 4 2 34 2 31 23 2 13 2 13 2 = + += + = x x xx x xx x3.32 Sol: (the coefficient matrix is the matrix in Ex 3) A A = x b(a) 0 (from Ex 3) A =

This system does not have a unique solution (b) (the coefficient matrix is the matrix in Ex 3) B B = x b12 0 (from Ex 3) B = = This system has a unique solution