SECTION 11.4 Matrix Algebra 807 5 Solve a System of Linear Equations Using an Inverse Matrix Inverse matrices can be used to solve systems of equations in which the number of equations is the same as the number of variables. Using the Inverse Matrix to Solve a System of Linear Equations Solve the system of equations: + = − + + =− + = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y x y z y z 3 3 4 3 4 3 2 Solution EXAMPLE 15 Let = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A X x y z B 1 1 0 1 3 4 0 4 3 3 3 2 Then the original system of equations can be written compactly as the matrix equation AX B = (4) From Example 13, the matrix A has the inverse A .1− Multiply each side of equation (4) by A .1− ( ) ( ) = = = = = − − − − − − AX B A AX A B A A X A B I X A B X A B 1 1 1 1 3 1 1 (5) Now use (5) to find = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ X x y z . This can be done either algebraically or with a graphing utility. Multiply both sides by −A .1 Associative Property of multiplication Definition of an inverse matrix Property of the identity matrix Algebraic Solution = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = = − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ − X x y z A B 7 4 3 4 1 3 4 3 4 1 1 1 1 3 3 2 1 2 2 1 ↑ Example 13 Graphing Solution Enter the matrices A and B into a graphing utility. Figure 18 shows the solution to the system of equations using the Desmos Matrix Calculator. = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ − A B A B 1 1 0 1 3 4 0 4 3 3 3 2 1 2 2 1 Figure 18 Desmos Matrix Calculator The solution is x y z 1, 2, 2 = = = − or, using an ordered triplet, 1, 2, 2 . ( ) − The method used in Example 15 to solve a system of equations is particularly useful when it is necessary to solve several systems of equations in which the constants appearing to the right of the equal signs change, but the coefficients of the variables on the left side remain the same. See Problems 45–64 for some illustrations. Be careful; this method can be used only if the inverse exists. If the matrix of the coefficients is singular, a different method must be used and the system is either inconsistent or dependent. Now Work PROBLEM 49
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