## Problem on the number of solutions to an inhomogeneous linear system

How many solutions are there to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}?$$
• ## Solution

We observe that this equation is a linear system of the form $\bfA \bfx = \bfb$ with nonzero $\bfb$. Hence, it is an inhomogeneous linear system.
Recall that
Identifying $\bfA$ as the matrix in the problem and computing its determinant, we have
Hence, there are either no solutions or there are infinitely many of them.
We try to find a single solution to $$\begin{pmatrix}1&1&1\\1&1&0\\0&0&1\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}$$ by writing out each of the three equations that it corresponds to:
\begin{align}
x + y + z &= 1\\
x + y \phantom{+ z }&= 2\\
z &=3
\end{align}
We plug the value of $z$ into the first equation to see \begin{align}x+y &=-2\\x+y&=\phantom{-}2 \end{align}
These equations are a contradiction. Thus, there are no solutions to the linear system.