Summary: Solving Systems With Inverses

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	${A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],\text{ where }ad-bc\ne 0$

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.


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