GUIDED PRACTICE for Example – – 2 12 – 4 – 6 A = Use a graphing calculator to find the inverse of the matrix A. Check the result by showing that AA -1 = I and A -1 A = I.

EXAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x – 3y = 19 x + 4y = – 7 Equation 1 Equation 2 SOLUTION STEP 1 Write the linear system as a matrix equation AX = B. coefficient matrix of matrix of matrix (A) variables (X) constants (B) 2 – xyxy 19 – 7 =

EXAMPLE 4 Solve a linear system STEP 2 Find the inverse of matrix A. 4 3 – 1 2 = A –1 = 1 8 – (–3) – STEP 3 Multiply the matrix of constants by A –1 on the left. X = A –1 B = – 2 19 – 7 = 5 – 3 = xyxy

EXAMPLE 4 Solve a linear system The solution of the system is (5, – 3). ANSWER CHECK 2(5) – 3(–3) = = (–3) = 5 – 12 = – 7

GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the linear system. 4x + y = 10 3x + 5y = – SOLUTION STEP 1 Write the linear system as a matrix equation AX = B. coefficient matrix of matrix of matrix (A) variables (X) constants (B) xyxy 10 – 1 =

GUIDED PRACTICE for Examples 4 and 5 STEP 2 Find the inverse of matrix A. STEP 3 Multiply the matrix of constants by A –1 on the left. 5 – 1 – 3 4 = A –1 = 1 20 – – – 3 –2 = X = A –1 B = – 4 10 – 1 = xyxy –

GUIDED PRACTICE for Examples 4 and 5 CHECK 4(3) + (–2) = 10 = 12 – 2 = 10 = 10 =10 3x + 5y = –1 3(3) + 5(– y) = –1 9 – 10 = –1 – 1 = –1 The solution of the system is (3, –2). ANSWER

GUIDED PRACTICE for Examples 4 and x – y = – 6 6x – 3y = – 18 SOLUTION STEP 1 Write the linear system as a matrix equation AX = B. coefficient matrix of matrix of matrix (A) variables (X) constants (B) 2 – 1 6 – 3. – x – y – 6 – 18 =

GUIDED PRACTICE for Examples 4 and 5 STEP 2 Find the inverse of matrix A. 3 1 – 6 2 = A –1 = –3 0 The inverse of matrix A is 0 so has infinitely many solutions. ANSWER