4-5 Matrix Inverses and Solving Systems Warm Up Lesson Presentation

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Presentation transcript:

4-5 Matrix Inverses and Solving Systems Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Multiple the matrices. 1. Find the determinant. 2. 3. –1

Objectives Determine whether a matrix has an inverse. Solve systems of equations using inverse matrices.

The identity matrix I has 1’s on the main diagonal and 0’s everywhere else. Remember!

Example 1A: Determining Whether Two Matrices Are Inverses by using your graphing calculator Determine whether the two given matrices are inverses. The product is the identity matrix I, so the matrices are inverses.

Example 1B: Determining Whether Two Matrices Are Inverses by using your graphing calculator Determine whether the two given matrices are inverses. Neither product is I, so the matrices are not inverses.

Check It Out! Example 1 Determine whether the given matrices are inverses. The product is the identity matrix I, so the matrices are inverses.

Example 2: Finding the Inverse of a Matrix by using your graphing calculator Find the inverse of the matrix if it is defined.

Check It Out! Example 3 Find the inverse of , if it is defined.

You can use the inverse of a matrix to solve a system of equations You can use the inverse of a matrix to solve a system of equations. This process is similar to solving an equation such as 5x = 20 by multiplying each side by , the multiplicative inverse of 5. To solve systems of equations with the inverse, you first write the matrix equation AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

The matrix equation representing is shown.

To solve AX = B, multiply both sides by the inverse A-1. A-1AX = A-1B IX = A-1B The product of A-1 and A is I. X = A-1B

Example 4: Solving Systems Using Inverse Matrices Write the matrix equation for the system and solve. Step 1 Set up the matrix equation. A X = B Write: coefficient matrix  variable matrix = constant matrix.

Step 2: Solve by plugging [A]-1 B in to your graphing calculator Example 4 Continued Step 2: Solve by plugging [A]-1 B in to your graphing calculator Solution is .

Example 5 Write the matrix equation for and solve.

Lesson Quiz: Part I 1. Determine whether and are inverses. yes 2. Find the inverse of , if it exists.

Lesson Quiz: Part II Write the matrix equation and solve. 3. 4. Decode using . "Find the inverse."