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Four.III Laplace’s Expansion Linear Algebra Jim Hefferon
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Laplace’s formula for the determinant
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1.1 Example Consider the permutation expansion. t 1,1 t 1,2 t 1,3 t 2,1 t 2,2 t 2,3 t 3,1 t 3,2 t 3,3 = t 1,1 t 2,2 t 3,3 1 0 0 0 1 0 0 0 1 + t 1,1 t 2,3 t 3,2 1 0 0 0 0 1 0 1 0 + t 1,2 t 2,1 t 3,3 0 1 0 1 0 0 0 0 1 + t 1,2 t 2,3 t 3,1 0 1 0 0 0 1 1 0 0 + t 1,3 t 2,1 t 3,2 0 0 1 1 0 0 0 1 0 + t 1,3 t 2,2 t 3,1 0 0 1 0 1 0 1 0 0 Pick a row or column and factor out its entries; here we do the entries in the first row.
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= t 1,1 · t 2,2 t 3,3 1 0 0 0 1 0 0 0 1 + t 2,3 t 3,2 1 0 0 0 0 1 0 1 0 + t 1,2 · t 2,1 t 3,3 0 1 0 1 0 0 0 0 1 + t 2,3 t 3,1 0 1 0 0 0 1 1 0 0 + t 1,3 · t 2,1 t 3,2 0 0 1 1 0 0 0 1 0 + t 2,2 t 3,1 0 0 1 0 1 0 1 0 0 In those permutation matrices, swap to get the first rows into place. This requires one swap to each of the permutation matrices on the second line, and two swaps to each on the third line. (Recall that row swaps change the sign of the determinant.)
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= t 1,1 · t 2,2 t 3,3 1 0 0 0 1 0 0 0 1 + t 2,3 t 3,2 1 0 0 0 0 1 0 1 0 - t 1,2 · t 2,1 t 3,3 1 0 0 0 1 0 0 0 1 + t 2,3 t 3,1 1 0 0 0 0 1 0 1 0 + t 1,3 · t 2,1 t 3,2 1 0 0 0 1 0 0 0 1 + t 2,2 t 3,1 1 0 0 0 0 1 0 1 0
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= t 1,1 · t 2,2 t 3,3 1 0 0 0 1 0 0 0 1 + t 2,3 t 3,2 1 0 0 0 0 1 0 1 0 - t 1,2 · t 2,1 t 3,3 1 0 0 0 1 0 0 0 1 + t 2,3 t 3,1 1 0 0 0 0 1 0 1 0 + t 1,3 · t 2,1 t 3,2 1 0 0 0 1 0 0 0 1 + t 2,2 t 3,1 1 0 0 0 0 1 0 1 0 On each line the terms in square brackets involve only the second and third row and column, and simplify to a 2 × 2 determinant. = t 1,1 · t 2,2 t 2,3 t 3,2 t 3,3 - t 1,2 · t 2,1 t 2,3 t 3,1 t 3,3 + t 1,3 · t 2,1 t 2,2 t 3,1 t 3,2
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Minor 1.2 Definition For any n × n matrix T , the ( n - 1 ) × ( n - 1 ) matrix formed by deleting row i
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