# Summary: Solve Systems With Inverses

## Key Equations

Identity matrix for a [latex]2\text{}\times \text{}2[/latex] matrix | [latex]{I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right][/latex] |

Identity matrix for a [latex]\text{3}\text{}\times \text{}3[/latex] matrix | [latex]{I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex] |

Multiplicative inverse of a [latex]2\text{}\times \text{}2[/latex] matrix | [latex]{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],\text{ where }ad-bc\ne 0[/latex] |

## Key Concepts

- An identity matrix has the property [latex]AI=IA=A[/latex].
- An invertible matrix has the property [latex]A{A}^{-1}={A}^{-1}A=I[/latex].
- Use matrix multiplication and the identity to find the inverse of a [latex]2\times 2[/latex] matrix.
- The multiplicative inverse can be found using a formula.
- Another method of finding the inverse is by augmenting with the identity.
- We can augment a [latex]3\times 3[/latex] matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse.
- Write the system of equations as [latex]AX=B[/latex], and multiply both sides by the inverse of [latex]A:{A}^{-1}AX={A}^{-1}B[/latex].
- We can also use a calculator to solve a system of equations with matrix inverses.

## Glossary

**identity matrix**a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra

**multiplicative inverse of a matrix**a matrix that, when multiplied by the original, equals the identity matrix

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