## Problem on solving a 2x2 matrix equation

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

We observe that this matrix equation is in the standard form $\bfA \bfx = \bfb$, where $\bfA = \begin{pmatrix}2 & 3 \\ 10 & 16\end{pmatrix}$, $\bfx = \begin{pmatrix}x\\y\end{pmatrix}$, and $\bfb = \begin{pmatrix}1\\2\end{pmatrix}.$
Recall that
Hence, we need to find the inverse of $\bfA$.
Recall that
Hence $$\bfA^{-1} = \begin{pmatrix} 2 & 3 \\ 10 & 16 \end{pmatrix}^{-1} = \frac{1}{2} \begin{pmatrix} 16 & -3 \\ -10 & 2 \end{pmatrix} = \begin{pmatrix} 8 & -3/2 \\ -5 & 1 \end{pmatrix}$$
The solution to the linear system is given by $\bfx = \bfA^{-1} \bfb$: $$\bfx = \begin{pmatrix} 8 & -3/2 \\ -5 & 1 \end{pmatrix} \begin{pmatrix} 1 \\2 \end{pmatrix}$$