4-7 Inverse Matrices & Systems

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Presentation transcript:

4-7 Inverse Matrices & Systems

Objectives Solving Systems of Equations Using Inverse Matrices

You can represent a system of equations with a matrix equation. Vocabulary You can represent a system of equations with a matrix equation. System of Equations Matrix Equation x + 2y = 5 3x + 5y = 14 Constant Matrix B Coefficient Matrix A Variable Matrix X

Writing a System as a Matrix Equation –3x – 4y + 5z = 11 –2x + 7y = –6 –5x + y – z = 20 Write the system as a matrix equation. Then identify the coefficient matrix, the variable matrix, and the constant matrix. Matrix equation: = –3 –4 5 –2 7 0 –5 1 –1 x y z 11 –6 20 Coefficient matrix –3 –4 5 –2 7 0 –5 1 –1 x y z Variable matrix 11 –6 20 Constant matrix

Solving a System of Two Equations Solve the system. 2x + 3y = –1 x – y = 12 2 3 1 –1 x y –1 12 = Write the system as a matrix equation. A–1 = Find A–1. 1 5 3 2 – = A–1B = = Solve for the variable matrix. x y 1 5 3 2 – –1 12 7 –5

Continued (continued) The solution of the system is (7, –5). Check: 2x + 3y = –1 x – y = 12 Use the original equations. 2(7) + 3(–5) –1 (7) – (–5) 12 Substitute. 14 – 15 = –1 7 + 5 = 12 Simplify.

Solving a System of Three Equations 7x + 3y + 2z = 13 –2x + y – 8z = 26 x – 4y +10z = –13 Solve the system . Step 1: Write the system as a matrix equation. Step 2: Store the coefficient matrix as matrix A and the constant matrix as matrix B. 7 3 2 –2 1 –8 1 –4 10 13 26 –13 x y z = The solution is (9, –12, –7).

Homework Pg 217 # 1, 2, 7, 8, 13