How Do I Find The Inverse Of A 2 x 2 Matrix?

Suppose the given square matrix is called A. To find the inverse of any matrix, we write the matrix in a larger matrix along side an identity matrix of the same size,

$\displaystyle \left[ \left. A\, \right|\,I \right]$

Now use row operations to rewrite this matrix so that the identity appears on the left side. The inverse of the original matrix will be on the right side of the transformed matrix,

$\displaystyle \left[ \left. I\, \right|\,{{A}^{-1}} \right]$

For instance, suppose we want to find the inverse of

$\displaystyle A=\left[ \begin{matrix} 2 & 2 \\ 2 & 1 \\ \end{matrix} \right]$

$\displaystyle \left[ \left. \begin{matrix} 2 & 2 \\ 2 & 1 \\ \end{matrix}\, \right|\,\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$

$\displaystyle \frac{1}{2}{{R}_{1}}\to {{R}_{1}}$

$\displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 2 & 1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ 0 & 1 \\ \end{matrix} \right]$

$\displaystyle -2{{R}_{1}}+{{R}_{2}}\to {{R}_{2}}$

$\displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 0 & -1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ -1 & 1 \\ \end{matrix} \right]$

$\displaystyle -1{{R}_{2}}\to {{R}_{2}}$

$\displaystyle \left[ \left. \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right|\begin{matrix} \frac{1}{2} & 0 \\ 1 & -1 \\ \end{matrix} \right]$

$\displaystyle -1{{R}_{2}}+{{R}_{1}}\to {{R}_{1}}$

$\displaystyle \left[ \left. \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right|\begin{matrix} -\frac{1}{2} & 1 \\ 1 & -1 \\ \end{matrix} \right]$

Let’s apply this strategy to finding a few more inverses.

Problem 1 Find the inverse of

$\displaystyle \left[ \begin{matrix} 2 & 4 \\ 2 & 5 \\ \end{matrix} \right]$

Problem 2 Find the inverse of

$\displaystyle \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \\ \end{matrix} \right]$