## Key Equations

 Identity matrix for a $2\text{}\times \text{}2$ matrix ${I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ Identity matrix for a $\text{3}\text{}\times \text{}3$ matrix ${I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ Multiplicative inverse of a $2\text{}\times \text{}2$ matrix ${A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],\text{ where }ad-bc\ne 0$

## Key Concepts

• An identity matrix has the property $AI=IA=A$.
• An invertible matrix has the property $A{A}^{-1}={A}^{-1}A=I$.
• Use matrix multiplication and the identity to find the inverse of a $2\times 2$ 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 $3\times 3$ 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 $AX=B$, and multiply both sides by the inverse of $A:{A}^{-1}AX={A}^{-1}B$.
• 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