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4.7 Inverse Matrices and Systems

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1) Inverse Matrices and Systems of Equations You have solved systems of equations using graphing, substitution, elimination…oh my… In the “real world”, these methods take too long and are almost never used. Inverse matrices are more practical.

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1) Inverse Matrices and Systems of Equations For a System of Equations

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1) Inverse Matrices and Systems of Equations For a We can write a System of Equations Matrix Equation

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1) Inverse Matrices and Systems of Equations Example 1: Write the system as a matrix equation

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1) Inverse Matrices and Systems of Equations Example 1: Write the system as a matrix equation Matrix Equation

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1) Inverse Matrices and Systems of Equations Example 1: Write the system as a matrix equation Matrix Equation Coefficient matrix Constant matrix Variable matrix

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1) Inverse Matrices and Systems of Equations Example 2:

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1) Inverse Matrices and Systems of Equations Example 2:

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1) Inverse Matrices and Systems of Equations Example 2: ABX

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1) Inverse Matrices and Systems of Equations

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When rearranging, take the inverse of A

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1) Inverse Matrices and Systems of Equations The Plan… “Solve the system” using matrices and inverses

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 1: Write a matrix equation

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 1: Write a matrix equation

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 2: Find the determinant and A -1

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 2: Find the determinant and A -1 Change signs Change places

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 2: Find the determinant and A -1 Change signs Change places detA = 4 – 3 = 1

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 2: Find the determinant and A -1

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 3: Solve for the variable matrix

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 3: Solve for the variable matrix

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 3: Solve for the variable matrix

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1) Inverse Matrices and Systems of Equations Example 3: Solve the system Step 3: Solve for the variable matrix The solution to the system is (4, 1).

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer.

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer.

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer. detA = 10 - 9 = 1

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer.

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer. The solution to the system is (-1, 4).

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1) Inverse Matrices and Systems of Equations Example 4: Solve the system. Check your answer. Check

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What about a matrix that has no inverse? It will have no unique solution. 1) Inverse Matrices and Systems of Equations

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Example 5: Determine whether the system has a unique solution.

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1) Inverse Matrices and Systems of Equations Example 5: Determine whether the system has a unique solution. Find the determinant.

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1) Inverse Matrices and Systems of Equations Example 5: Determine whether the system has a unique solution. Find the determinant.

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1) Inverse Matrices and Systems of Equations Example 5: Determine whether the system has a unique solution. Find the determinant. Since detA = 0, there is no inverse. The system does not have a unique solution.

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Homework p.217 #1-5, 7-10, 20, 21, 23, 24, 26, 27, 36 DUE TOMORROW: Two codes TEST: Wednesday Nov 25 Chapter 4

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