2.7 - Section 2.7 Determinants Definition: The determinant...

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Section 2.7 Determinants Definition: The determinant of a 2X2 matrix ab A cd  =   is det(A) = = ad - bc Example: Find 32 06 Solution : -3(6) - 0 (2) = -18
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Definition : The ijth minor , M ij , is found by deleting the ith row and jth column of matrix A. Example: 123 456 789 A   =  (i) Find M 13 13 45 78 M = (ii) Find M 32 32 13 46 M = Definition: The ijth cofactor , A ij , is found by (1 ) d e t ( ) ij ij ij A M + =−
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Example: 123 456 789 A = (i)Find A 13 . () 13 13 45 1 78 A + =− (1) 4(8) 7(5) 32 35 3 =−= (i)Find A 32 . 32 32 (1 ) 46 A + ( 1) 1(6) 4(3) ) ( 6 1 2 ) 6 = Definition: Cofactor Expansion Cofactor expansion along row i a i1 A i1 +a i2 A i2 +...+a in A in
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Cofactor expansion along column j a 1j A 1j +a 2j A 2j +...+a nj A nj Theorem: Cofactor expansion along any row = Cofactor expansion along any column. Definition: The determinant of a matrix is the value found from cofactor expansion. Example: 123 456 789 Pick any row OR any column. For row 1, the cofactor expansion is a 11 A 11 +a 12 A 12 +a 13 A 13 = 11 56 1( 1) 89 + 12 46 2( 1) 79 + +− 13 45 3( 1) 78 +
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=(5(9)-6(8)) + (-2) (4(9)-7(6)) + 3 (4(8)-7(5)) =(45-48) + (-2) (36 - 42) + 3 (32 - 35)
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2.7 - Section 2.7 Determinants Definition: The determinant...

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