lecture_12 - MA 265 LECTURE NOTES: MONDAY, FEBRUARY 4...

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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, FEBRUARY 4 Determinants Review. In the previous lecture, we defined the determinant of an n n matrix as the following number: a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a n 1 a n 2 a nn = X S n ( ) a 1 (1) a 2 (2) a n ( n ) as a sum over the n ! permutations , where ( ) = 1 depending on whether is even or odd. As specific examples, we showed that det[ a ] = a and a b c d = ad- bc. (Be careful not to confuse this with absolute value!) Example. We compute the determinant of a 2 2 matrix: 2- 3 4 5 = (2)(5)- (- 3)(4) = 22 . Visually, we compute the determinant using the following arrows: ? ? ? ? ? ? ? ? ? ? ? ? ? ? a b c d The arrows mean multiply all of these numbers together, where right-ward arrows correspond to + and the left-ward arrows correspond to - . This diagram says +( a d )- ( b c ). 3 3 Determinants. We show how to compute the determinant of a 3 3 matrix. Recall that we have 3! = 6 permutations to consider: 1 : 1 7 1 2 7 2 3 7 3 2 : 1 7 2 2 7 3 3 7 1 3 : 1 7 3 2 7 1 3 7 2 4 : 1 7 1 2 7 3 3 7 2 5 : 1 7 2 2 7 1 3 7 3 6 : 1 7 3 2 7 2 3 7 1 We showed in the previous lecture that the first three have signs ( 1 ) = ( 2 ) = ( 3 ) = +1, and the last three have signs ( 4 ) = ( 5 ) = ( 6 ) =- 1. Using the definition of the determinant, the 3 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 2 MA 265 LECTURE NOTES: MONDAY, FEBRUARY 4 has determinant det( A ) = ( 1 ) a 11 a 22 a 33 + ( 2 ) a 12 a 23 a 31 + ( 3 ) a 13 a 21 a 32 + ( 4 ) a 11 a 23 a 32 + ( 5 ) a 12 a 21 a 33 + ( 6 ) a 13 a 22 a 31 = ( a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 )- ( a 11 a 23 a 32 + a 12 a 21 a 33 + a 13 a 22 a 31 ) ....
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lecture_12 - MA 265 LECTURE NOTES: MONDAY, FEBRUARY 4...

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