SECTION 11.3 Systems of Linear Equations: Determinants 787 If the determinant D of the coefficients of the variables equals 0 (so that Cramer’s Rule cannot be used), then the system either is consistent with dependent equations or is inconsistent.To determine whether the system has no solution or infinitely many solutions, solve the system using the methods of Section 11.1 or Section 11.2. Now Work PROBLEM 15 3 Evaluate 3 by 3 Determinants To use Cramer's Rule to solve a system of three equations containing three variables, we need to define a 3 by 3 determinant. A 3 by 3 determinant is symbolized by a a a a a a a a a 11 12 13 21 22 23 31 32 33 (8) in which … a a a , , , 11 12 33 are real numbers. As with matrices, we use a double subscript to identify an entry by indicating its row and column numbers. For example, the entry a23 is in row 2, column 3. The value of a 3 by 3 determinant may be defined in terms of 2 by 2 determinants by the following formula: = − + a a a a a a a a a a a a a a a a a a a a a a a a 11 12 13 21 22 23 31 32 33 11 22 23 32 33 12 21 23 31 33 13 21 22 31 32 (9) Algebraic Solution Enter the coefficient matrix into the graphing utility. Call it A and evaluate [ ]A det . Since [ ] ≠ A det 0, Cramer’s Rule can be used. Enter the matrices Dx and Dy into the graphing utility and call them B and C, respectively. Finally, find x by calculating [ ] [ ] B A det det , and find y by calculating [ ] [ ] C A det det . The results are shown in Figure 11 using the Desmos Matrix Calculator. The determinant D of the coefficients of the variables is ( ) = − = ⋅ − − = D 3 2 6 1 3 1 6 2 15 Because ≠ D 0, Cramer’s Rule (7) can be used. ( ) = = − = ⋅ − − = = = = = ⋅ − ⋅ = = x D D y D D 4 2 13 1 15 4113 2 15 30 15 2 3 4 6 13 15 3 13 6 4 15 15 15 1 x y Graphing Solution Figure 11 The solution is = = x y 2, 1 or, using an ordered pair, ( ) 2, 1 . Minus ↓ Plus ↓ ↑ 2 by 2 determinant left after removing the row and column containing a11 ↑ 2 by 2 determinant left after removing the row and column containing a12 ↑ 2 by 2 determinant left after removing the row and column containing a13 The 2 by 2 determinants in formula (9) are called minors of the 3 by 3 determinant. For an n by n determinant, the minor Mij of entry aij is the ( ) −n 1 by ( ) −n 1 determinant that results from removing the ith row and the jth column.
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