Finding Matrix Inverses

Algorithm: Matrix Inversion

To find the inverse of an invertible matrix $A\in F^{n\times n}$:

1. Notate an $n\times 2n$-matrix $(A\mid I_n)$ by sticking the identity matrix $I_n$ to the right of $A$.

2. Perform Gauss-Jordan elimination on $(A\mid I_n)$. If $A$ is indeed invertible, the final result will be $(I_n\mid A^{-1})$.

EXAMPLE

Let’s find the inverse of $A=\left[\begin{array}{ccc}6& 8& 3\\ 4& 7& 3\\ 1& 2& 1\end{array}\right]$. Notate

$$[A\mid I_3]=\left[\begin{array}{cccccc}6& 8& 3& 1& 0& 0\\ 4& 7& 3& 0& 1& 0\\ 1& 2& 1& 0& 0& 1\end{array}\right]$$

Perform Gauss-Jordan elimination

$$\left[\begin{array}{cccccc}6& 8& 3& 1& 0& 0\\ 4& 7& 3& 0& 1& 0\\ 1& 2& 1& 0& 0& 1\end{array}\right]\approx \left[\begin{array}{cccccc}1& 2& 1& 0& 0& 1\\ 6& 8& 3& 1& 0& 0\\ 4& 7& 3& 0& 1& 0\end{array}\right]\approx \left[\begin{array}{cccccc}1& 2& 1& 0& 0& 1\\ 0& -4& -3& 1& 0& -6\\ 0& -1& -1& 0& 1& -4\end{array}\right]$$ $$\left[\begin{array}{cccccc}1& 2& 1& 0& 0& 1\\ 0& -4& -3& 1& 0& -6\\ 0& -1& -1& 0& 1& -4\end{array}\right]\approx \left[\begin{array}{cccccc}1& 0& -1& 0& 2& -7\\ 0& 1& 1& 0& -1& 4\\ 0& 0& 1& 1& -4& 10\end{array}\right]\approx \left[\begin{array}{cccccc}1& 0& 0& 1& -2& 3\\ 0& 1& 0& -1& 3& -6\\ 0& 0& 1& 1& -4& 10\end{array}\right]$$ $$A^{-1}=\left[\begin{array}{ccc}1& -2& 3\\ -1& 3& -6\\ 1& -4& 10\end{array}\right]$$

Theorem: Inverting $2\times 2$-Matrices

A $2\times 2$-matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is invertible if and only if

$$ad-bc\ne 0$$

If $A$ is invertible, then

$$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

PROOF

TODO