14.3 Matrix Equations and Matrix Solutions to 2x2 Systems OBJ: Use the Inverse of a 2 x 2 Matrix to solve a system of equations

To find the inverse (A -1 ) of Matrix A EX:  If A =  54   23  Find A -1  A  5 3 – 4 2 A -1 = 1  3-4  7  -2 5  Check your answer by finding A A -1 1  5 4   3-4  7  2 3   -2 5  If A =  ab   cd  where  A  = ad – bc then A -1 = 1  d -b   A   -c a 

Use matrices to solve the given system of equations: 5x + 4y = -2 2x + 3y = -5 This system of two equations in two unknowns can be replaced by a single matrix equation. A X = C  5 4   x   -2   2 3   y  =  -5 

Notice that the coefficient matrix: A =  5 4   2 3  is the matrix whose inverse was found in the previous example. A -1 = 1  3 -4  7  -2 5 

If you left-multiply both sides of the matrix equation by A -1, you get: 1  3 -4   5 4   x  1  3 -4   -2  7  -2 5   2 3   y  = 7  -2 5   -5   7 0   x  1  14  7  0 7   y  = 7  -21   1 0   x  =  2   0 1   y   -3   x  =  2   y   -3 

Solve the matrix equation for X.  7 -2   3 -5   -1 5   -4 1  X –  0 4  =  2 -3  = +  7 -2   2 0   -4 1  X =  2 1  If you left-multiply both sides of the matrix equation by A -1 1  1 2   7 -2  1  1 2   2 0  -1  4 7   -4 1  X = -1  4 7   2 1   -1 0  1  6 2  -1  0 -1  X = -1  22 7    X =  