Summary: Solving 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