907 9.3 Determinant Solution of Linear Systems 9.3 Determinant Solution of Linear Systems ■ Determinants ■ Cofactors ■ n : n Determinants ■ Determinant Theorems ■ Cramer’s Rule Determinants Every n * n matrix A is associated with a real number called the determinant of A, written ∣ A∣. In this section we show how to evaluate determinants of square matrices, providing mathematical justification as we proceed. Graphing calculators are programmed to evaluate determinants in their matrix menus. The determinant of a 2 * 2 matrix is defined as follows. Determinant of a 2 : 2 Matrix If A = c a11 a21 a12 a22d , then ∣ A∣ = ` a11 a21 a12 a22 ` =a11a22 −a21a12. NOTE Matrices are enclosed with square brackets, while determinants are denoted with vertical bars. A matrix is an array of numbers, but its determinant is a single number. EXAMPLE 1 Evaluating a 2 : 2 Determinant Let A = c -3 6 4 8d . Find A . GRAPHING CALCULATOR SOLUTION We can define a matrix and then use the capability of a graphing calculator to find the determinant of the matrix. In the screen in Figure 9, the symbol det13A42 represents the determinant of 3A4. Figure 9 S Now Try Exercise 7. ALGEBRAIC SOLUTION Use the definition with a11 = -3, a12 = 4, a21 = 6, a22 = 8. 0 A0 = ` -3 4 6 8 ` A = -3 # 8 - 6 # 4 a11 a22 a21 a12 = -24 - 24 Multiply. = -48 Subtract. The arrows in the diagram below indicate which products to find when evaluating a 2 * 2 determinant. 2 a11 a12 a21 a22 2
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