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# Introduction to Solving Systems with Inverses

### Learning Outcomes

By the end of this section, you will be able to:
• Verify that multiplying a matrix by its inverse results in 1.
• Use matrix multiplication to find the inverse of a matrix.
• Find an inverse by augmenting with an identity matrix.
• Solve a 2x2 system using an inverse.
• Solve a 3x3 systems using an inverse.
• Solve a system with a calculator.
Nancy plans to invest \$10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Nancy invest in each bond? What is the best method to solve this problem? There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.