982 CHAPTER 9 Systems and Matrices Concepts Examples Scalar Multiplication To multiply a matrix by a scalar, multiply each element of the matrix by the scalar. Matrix Multiplication The product AB of an m* n matrix A and an n * p matrix B is found as follows. To find the i th row, j th column element of matrix AB, multiply each element in the ith row of A by the corresponding element in the jth column of B. The sum of these products will give the row i, column j element of AB. Find the product. 3£ 6 1 0 2 -2 8§ = £ 18 3 0 6 -6 24§ Multiply each element by the scalar 3. Find the matrix product. £ 1 5 -8 -2 0 7 3 4 -7§ £ 1 -2 3§ = £ 1112 + 1-221-22 + 3132 5112 + 01-22 + 4132 -8112 + 71-22 + 1-72132 § 3 * 3 3 * 1 = £ 14 17 -43§ 3 * 1 9.8 Matrix Inverses Finding an Inverse Matrix To obtain A-1 for any n * n matrix A for which A-1 exists, follow these steps. Step 1 Form the augmented matrix 3A In4, where In is the n * n identity matrix. Step 2 Perform row transformations on 3A In4 to obtain a matrix of the form 3In B4. Step 3 Matrix B is A-1. Find A-1 if A = c 5 2 2 1d . c 5 2 2 1 2 1 0 0 1d c 1 2 0 1 2 1 0 -2 1d -2R2 + R1 c 1 0 0 1 2 1 -2 -2 5d -2R1 + R2 ()* (11)11* I2 A-1 Therefore, A-1 = c 1 -2 -2 5d . Chapter 9 Review Exercises Use the substitution or elimination method to solve each system of equations. Identify any inconsistent systems or systems with infinitely many solutions. If a system has infinitely many solutions, write the solution set with y arbitrary. 1. 2x + 6y = 6 5x + 9y = 9 2. 3x - 5y = 7 2x + 3y = 30 3. x + 5y = 9 2x + 10y = 18 4. 1 6 x + 1 3 y = 8 1 4 x + 1 2 y = 12 5. y = -x + 3 2x + 2y = 1 6. 0.2x + 0.5y = 6 0.4x + y = 9 7. 3x - 2y = 0 9x + 8y = 7 8. 6x + 10y = -11 9x + 6y = -3 9. 2x - 5y + 3z = -1 x + 4y - 2z = 9 -x + 2y + 4z = 5
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