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

Lesson Quiz Holt Algebra 2

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

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**Objectives Determine whether a matrix has an inverse.**

Solve systems of equations using inverse matrices.

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

Remember!

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**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.

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**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.

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

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**Example 2: Finding the Inverse of a Matrix by using your graphing calculator**

Find the inverse of the matrix if it is defined.

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

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**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.

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**The matrix equation representing is shown.**

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**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

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**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.

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**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 .

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Example 5 Write the matrix equation for and solve.

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

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

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