Matrix inverses

The matrix $\bfB$ is the matrix inverse of $\bfA$ if $\bfA \bfB = \bfI$ and $\bfB \bfA = \bfI.$

The matrix inverse of $\bfA$ is denoted $\bfA^{-1}$.

The inverse of a $2\times2$ matrix is given by swapping the diagonal entries, negating the off-diagonal entries, and dividing by the determinant: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$$

To find the inverse of a $3 \times 3$ matrix,

Compute the minors of each element
Negate every other element, according to a checkerboard pattern
Take the transpose
Divide by the determinant of the original matrix

Study Guide

Inverse of a 2x2 matrix
(1 problem)
Inverse of a 3x3 matrix
(1 problem)

Practice problems

If $\bfA$ is an invertible matrix, show that $\bigl(\bfA^t \bigr)^{-1} = \bigl( \bfA^{-1} \bigr) ^t$. That is, show that the inverse of the transpose is the transpose of the inverse.
Solution
Find $\bfA^{-1}$ and use it to solve $\bfA \bfx = \bfb$, where $$\bfA = \begin{pmatrix}1 & -1& 1 \\ 1 & 1 & 1\\1 & 2 & 4 \end{pmatrix}, \quad \bfb = \begin{pmatrix} 1\\3\\2 \end{pmatrix}$$
Solution
Solve by matrix inversion: $$\begin{pmatrix} 2 & 3 \\ 10 & 16 \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix}.$$
Solution

Matrix inverses

Study Guide

Related topics

Practice problems