Matrix Inverses

This is a guide to inverting \(1\times 1\), \(2\times 2\), and \(n\times n\) matrices.

Let \(A\) be the \(1\times 1\) matrix $$\begin{align} A &= \begin{bmatrix} a \end{bmatrix}. \end{align}$$ The inverse is simply the reciprocal: $$\begin{align} A^{-1} &= \begin{bmatrix} 1/a \end{bmatrix}. \end{align}$$

Let \(B\) be the \(2\times 2\) matrix $$\begin{align} B &= \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}. \end{align}$$ It can be shown that the inverse follows a simple pattern: $$\begin{align} B^{-1} &= \frac{1} {\det B} \begin{bmatrix} b_{22} & -b_{12} \\ -b_{21} & b_{11} \end{bmatrix} \\ &= \frac{1} {b_{11} b_{22} - b_{12} b_{21}} \begin{bmatrix} b_{22} & -b_{12} \\ -b_{21} & b_{11} \end{bmatrix}. \end{align}$$

Let \(C\) be an \(n\times n\) matrix. It can be shown that its inverse is $$\begin{align} C^{-1} = \frac{1} {\det C} \adj{C}, \end{align}$$ where \(\adj\) is the adjoint of \(C\).