971 9.8 Matrix Inverses A-11AX2 = A-1B Multiply each side on the left by A-1. 1A-1A2X = A-1B Associative property I3 X = A-1B Inverse property X = A-1B Identity property Matrix A-1B gives the solution of the system. Solution of the Matrix Equation AX=B Suppose A is an n * n matrix with inverse A-1, X is an n * 1 matrix of variables, and B is an n * 1 matrix. The matrix equation AX =B has the solution X =A−1B. This method of using matrix inverses to solve systems of equations is useful when the inverse is already known or when many systems of the form AX = B must be solved and only B changes. EXAMPLE 4 Solving Systems of Equations Using Matrix Inverses Solve each system using the inverse of the coefficient matrix. (a) 2x - 3y = 4 x + 5y = 2 (b) x + z = -1 2x - 2y - z = 5 3x = 6 ALGEBRAIC SOLUTION (a) The system can be written in matrix form as c 2 1 -3 5d c x yd = c 4 2d , where A = c 2 1 -3 5d , X = c x yd , and B = c 4 2d . An equivalent matrix equation is AX = B with solution X = A-1B. Use the methods described in this section to determine that A-1 = c 5 13 - 1 13 3 13 2 13d , and thus A-1B = c 5 13 - 1 13 3 13 2 13d c 4 2d = c 2 0d . X = A-1B, so X = c x yd = c 2 0d . The final matrix shows that the solution set of the system is 512, 026. GRAPHING CALCULATOR SOLUTION (a) Enter 3A4 and 3B4 as defined in the algebraic solution, and then find the product 3A4-13B4 as shown in Figure 30. The display indicates that the solution set is 512, 026. Notice that it is not necessary to actually compute 3A4-1 here. The calculator stores this inverse and then multiplies it by 3B4 to obtain the column matrix that represents the solution. Figure 30
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