910 CHAPTER 9 Systems and Matrices EXAMPLE 3 Evaluating a 3 : 3 Determinant Evaluate 3 2 -1 -1 -3 -4 0 -2 -3 23 , expanding by the second column. SOLUTION First find the minor of each element in the second column. Use parentheses, and keep track of all negative signs to avoid errors. M12 = 2 -1 -1 -3 22 = -1122 - 1-121-32 = -5 M22 = 2 2 -1 -2 22 = 2122 - 1-121-22 = 2 M32 = 2 2 -1 -2 -32 = 21-32 - 1-121-22 = -8 Now find the cofactor of each element of these minors. A12 = 1-121+2 # M12 = 1-123 # 1-52 = -11-52 = 5 A22 = 1-122+2 # M22 = 1-124 # 2 = 1 # 2 = 2 A32 = 1-123+2 # M32 = 1-125 # 1-82 = -11-82 = 8 Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products. 3 2 -1 -1 -3 -4 0 -2 -3 23 = a12 # A12 + a22 # A22 + a32 # A32 = -3152 + 1-422 + 0182 = -15 + 1-82 + 0 = -23 S Now Try Exercise 21. The matrix in Example 3 can be entered as [B] and its determinant found using a graphing calculator, as shown in the screen above. In Example 3, we would have found the same answer using any row or column of the matrix. One reason we used column 2 is that it contains a 0 element, so it was not really necessary to calculate M32 and A32. Instead of calculating 1-12i+j for a given element, we can use the sign checkerboard shown below. The signs alternate for each row and column, beginning with + in the first row, first column position. If we expand a 3 * 3 matrix about row 3, for example, the first minor would have a + sign associated with it, the second minor a - sign, and the third minor a + sign. Sign array for 3 * 3 matrices + - + - + - + - + This sign array can be extended for determinants of larger matrices.
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