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Study Guides > College Algebra CoRequisite Course

Summary: Solving Systems With Inverses

Key Equations

Identity matrix for a 2×22\text{}\times \text{}2 matrix I2=[1001]{I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]
Identity matrix for a 3×3\text{3}\text{}\times \text{}3 matrix I3=[100010001]{I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]
Multiplicative inverse of a 2×22\text{}\times \text{}2 matrix A−1=1ad−bc[d−b−ca], where ad−bc≠0{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=AAI=IA=A.
  • An invertible matrix has the property AA−1=A−1A=IA{A}^{-1}={A}^{-1}A=I.
  • Use matrix multiplication and the identity to find the inverse of a 2×22\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 3×33\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=BAX=B, and multiply both sides by the inverse of A:A−1AX=A−1BA:{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

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