SECTION 11.3 Systems of Linear Equations: Determinants 789 Observe that the signs, ( ) − + 1 , i j associated with the cofactors alternate between positive and negative. For example, for a 3 by 3 determinant, the signs follow the pattern − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 1 1 1 1 1 1 1 1 (10) It can be shown that the value of a determinant does not depend on the choice of the row or column used in the expansion. So, expanding across a row or column that has an entry equal to 0 reduces the amount of work needed to compute the value of the determinant. Evaluating a 3 by 3 Determinant Find the value of the 3 by 3 determinant: − − 3 0 4 4 6 2 8 2 3 Solution EXAMPLE 4 Because 0 is in row 1, column 2, it is easiest to expand across row 1 or down column 2. We choose to expand across row 1. For the signs of the cofactors, we use “ − 1, 1, 1” from row 1 of the 3 by 3 determinant in (10). ( ) ( ) ( ) [ ] ( )( ) ( )( ) − − = ⋅ ⋅ − + − ⋅ ⋅ + ⋅ − ⋅ − = − − + + − − − = ⋅ + − − = + = 3 0 4 4 6 2 8 2 3 1 3 6 2 2 3 1 0 4 2 8 3 1 4 4 6 8 2 3 18 4 0 4 8 48 3 22 4 56 66 224 290 4 Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables Consider the following system of three equations containing three variables. + + = + + = + + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ a x a y a z c a x a y a z c a x a y a z c 11 12 13 1 21 22 23 2 31 32 33 3 (11) It can be shown that if the determinant D of the coefficients of the variables is not 0, that is, if = ≠ D a a a a a a a a a 0 11 12 13 21 22 23 31 32 33 the system in (11) has a unique solution. Now Work PROBLEM 11 THEOREM Cramer’s Rule for Three Equations Containing Three Variables If ≠ D 0, the solution of the system in (11) is = = = x D D y D D z D D x y z where = = = D c a a c a a c a a D a c a a c a a c a D a a c a a c a a c x y z 1 12 13 2 22 23 3 32 33 11 1 13 21 2 23 31 3 33 11 12 1 21 22 2 31 32 3
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