# 15.5: Matrix Inversion

The inverse of a square matrix $$\mathbf{A}$$, sometimes called a reciprocal matrix, is a matrix $$\mathbf{A}^{-1}$$ such that $$\mathbf{A}\mathbf{A}^{-1}=\mathbf{I}$$, where $$\mathbf{I}$$ is the identity matrix.

It is easy to obtain $$\mathbf{A}^{-1}$$ in the case of a $$2\times 2$$ matrix:

$\mathbf{A}=\begin{pmatrix} a&b \\ c&d \end{pmatrix};\;\mathbf{A}^{-1}=\begin{pmatrix} e&f \\ g&h \end{pmatrix} \nonumber$

$\begin{pmatrix} a&b \\ c&d \end{pmatrix}\begin{pmatrix} e&f \\ g&h \end{pmatrix}=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} \nonumber$

$\label{eq:matrices_inverse1} ae+bg=1$

$\label{eq:matrices_inverse2} af+bh=0$

$\label{eq:matrices_inverse3} ce+dg=0$

$\label{eq:matrices_inverse4} cf+dh=1$

From Equations \ref{eq:matrices_inverse1} and \ref{eq:matrices_inverse3}: $$g=(1-ae)/b=-ce/d\rightarrow ae=cbe/d+1\rightarrow e\left(a-cb/d\right)=1\rightarrow e\left(ad-cb\right)=d\rightarrow e=d/(ad-cb)$$. You can obtain expressions for $$f,g$$ and $$h$$ in a similar way to obtain:

$\mathbf{A}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b \\ -c&a \end{pmatrix} \nonumber$

Notice that the term $$(ad-bc)$$ is the determinant of $$\mathbf{A}$$, and therefore $$\mathbf{A}^{-1}$$ exists only if $$|\mathbf{A}|\neq 0$$. In other words, the inverse of a singular matrix is not defined.

If you think about a square matrix as an operator, the inverse “undoes” what the original matrix does. For example, the matrix $$\begin{pmatrix} -2&0 \\ 0&1 \end{pmatrix}$$, when applied to a vector $$(x,y)$$, gives $$(-2x,y)$$:

$\begin{pmatrix} -2&0 \\ 0&1 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}=\begin{pmatrix} -2x\\ y \end{pmatrix} \nonumber$

The inverse of $$\mathbf{A}$$, when applied to $$(-2x,y)$$, gives back the original vector, $$(x,y)$$:

$\mathbf{A}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b \\ -c&a \end{pmatrix}\rightarrow \mathbf{A}^{-1}= -\frac{1}{2}\begin{pmatrix} 1&0 \\ 0&-2 \end{pmatrix} \nonumber$

$-\frac{1}{2}\begin{pmatrix} 1&0 \\ 0&-2 \end{pmatrix} \begin{pmatrix} -2x\\ y \end{pmatrix}=\begin{pmatrix} x\\ y \end{pmatrix} \nonumber$

It is of course possible to calculate the inverse of matrices of higher dimensions, but in this course you will not be required to do so by hand.