2331_Notes_5o2_fill - Math 2331 Linear Algebra Section 5.2...

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Math 2331 Linear Algebra Section 5.2 Determinants and Cofactors The properties for determinants in the last section are great, but don't really tell us HOW to find the determinant of a matrix if it is bigger than 2x2. This section develops formulas for determinants in several different ways. PIVOT FORMULA - If A is an nxn matrix, we know we can use elimination to factor A in the form PA = LU P is a permutation matrix L is lower triangular with 1's along the main diagonal U is the upper triangular form of A from elimination By the formula for products, det(P)*det(A) = det(L) * det(U) OR- Finding the determinant by reducing to upper triangular is definitely one way to find a determinant and often a good way. But, not the only way! Before I talk about permutations, let's look at row reducing a general 3x3 matrix - 11 12 13 21 22 23 31 32 33 aaa Aaa a ⎛⎞ ⎜⎟ = ⎝⎠ Assume the upper left corner is not 0, Multiply row 2 and 3 by the upper left corner entry-
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⎛⎞ ⎜⎟ ⎝⎠ aaa aa aa aa 11 12 13 11 21 11 22 11 23 11 31 11 32 11 33 Now, subtract to get 0's below the pivot in the first row first column −− aa a aa aa 11 12 13 11 22 12 21 11 23 13 21 11 32 31 12 11 33 31 13 0 0 Now, eliminate the entry in the middle of the bottom row Multiply row 3 by the A(2, 2) entry and subtract 11 12 13 11 22 12 21 11 23 13 21 11 32 31 12 11 22 12 21 11 33 31 13 11 22 12 21 0 0 ( )( ) ( )( ) a Subtract 11 32 31 12 3( ) 2 Row a a a a
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This note was uploaded on 08/10/2011 for the course MATH 2331 taught by Professor Staff during the Spring '08 term at University of Houston.

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2331_Notes_5o2_fill - Math 2331 Linear Algebra Section 5.2...

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