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lec3_3 - Example 5 Use cofactor expansion along the 3 rd...

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NOTES ON SECTION 3 . 3 MA265, SECTIONS 41, 52 Key words minor expansion along a row/column cofactor area Definition 1. If A is a n × n matrix, let M ij be the matrix obtained by deleting the i th row and j th column. The ij th minor of A (or the minor of a ij ) is the number det M ij . Example 2. If A = 1 2 3 4 5 6 7 8 9 then M 23 = 1 2 7 8 and the minor of a 23 is det M 23 = 8 - 14 = - 6 . Definition 3. The cofactor of a ij is A ij = ( - 1) i + j det M ij . The cofactor is the minor of a ij up to a sign error, and the sign error is determined by the following “checkerboard” rule: + - + . . . - + - . . . + - + . . . . . . . . . . . . . . . Theorem 4. Pick a row r . Then det A = n X j =1 a rj A rj = a r 1 A r 1 + a r 2 A r 2 + · · · + a rn A rn Or, pick a column c . Then det A = n X i =1 a ic A ic = a 1 c A 1 c + a 2 c A 2 c + · · · + a nc A nc See the textbook/class notes for various examples of calculations using this method.
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Unformatted text preview: Example 5. Use cofactor expansion along the 3 rd row to calculate det A where A = 1-2 1 2 1 2-1-1 1 2 1 1 Since a 31 = a 32 = a 34 = 0 there is only one non-zero term det A = 0 + 0 + (-1) · A 33 + 0 where A 33 = (+) · det 1-2 2 1-1 1 2 1 1 Since it’s a 3 × 3 determinant we can compute this fairly easily to be 1 + 2 + 0--(-2)-(-4) = 9, so the final answer is det A = a 33 A 33 = (-1) · 9 =-9 MATLAB agrees: >>> det([1,-2,1,0;2,1,2,-1;0,0,-1,0;1,2,1,1]) ans = -9 2...
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