919 9.3 Determinant Solution of Linear Systems Solve each system for x and y using Cramer’s rule. Assume a and b are nonzero constants. 105. bx + y = a2 ax + y = b2 106. ax + by = b a x + y = 1 b 107. b2x + a2y = b2 ax + by = a 108. x + by = b ax + y = a 109. Use Cramer’s rule to find the solution set if a, b, c, d, e, and ƒ are consecutive integers. ax + by = c dx + ey = ƒ 110. In the following system, a, b, c, c, l are consecutive integers. Express the solution set in terms of z. ax + by + cz = d ex + ƒy + gz = h ix + jy + kz = l Relating Concepts For individual or collaborative investigation (Exercises 111–114) The determinant of a 3 * 3 matrix A is defined as follows. If A = £ a11 a21 a31 a12 a22 a32 a13 a23 a33§, then A = 3 a11 a21 a31 a12 a22 a32 a13 a23 a33 3 = 1a11a22a33 + a12a23a31 + a13a21a322 - 1a31a22a13 + a32a23a11 + a33a21a122. Work these exercises in order. 111. The determinant of a 3 * 3 matrix can also be found using the method of “diagonals.” Step 1 Rewrite columns 1 and 2 of matrix A to the right of matrix A. Step 2 Identify the diagonals d1 through d6 and multiply their elements. Step 3 Find the sum of the products from d1, d2, and d3. Step 4 Subtract the sum of the products from d4, d5, and d6 from that sum: 1d1 + d2 + d32 - 1d4 + d5 + d62. Verify that this method produces the same results as the previous method given. £a11 a21 a31 a12 a22 a32 a13 a23 a33§ a11 a21 a31 a12 a22 a32 d4 d5 d6 d1 d2 d3 Each d is a product. 112. Evaluate the determinant 3 1 0 7 3 2 1 2 6 53 using the method of “diagonals.” 113. See Exercise 112. Evaluate the determinant by expanding about column 1 and using the method of cofactors. Do these methods give the same determinant for 3 * 3 matrices? 114. Concept Check Does the method of evaluating a determinant using “diagonals” extend to 4 * 4 matrices?
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