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USING THE INVERSE MATRIX METHOD to solve systems of equations The concept of solving systems of equations using inverse matrices is similar to the concept of solving simple equations. For example, to solve 7 x 14 , we multiply both sides by the multiplicative inverse of 7, which is 1/7: 1 1 7 x 14 7 7 On the left, (1/7) × 7 = 1. The number 1 is the “identity” for multiplication of ordinary numbers. The solution for our equation is: x2 We will extend this concept to solving systems of simultaneous equations. In general, if we want to solve a system of equations: Let Then ax by e cx dy f a b A c d x X y e B f Coefficient Matrix Variable Matrix Product Matrix AX B If we multiply each side by A-1, we have: However, we know that A1 A I , the identity matrix. So we obtain: But, IX X , so the solution to the system of equations is given by: Example: Solve 2 x 11y 25 3x 2 y 6 Solution: Write the system in the form of a matrix equation AX B . Multiply both sides of the matrix equation by A–1 and solve.