###### Exercise 5

Consider the system of equations described by the following augmented matrix.

\begin{equation*}
\left[
\begin{array}{ccc|c}
1 \amp 2 \amp 3 \amp 1 \\
4 \amp 5 \amp 6 \amp 4 \\
a \amp b \amp c \amp 9 \\
\end{array}
\right]\text{.}
\end{equation*}

Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to be inconsistent. Explain why your choice has this property.

Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to have a unique solution. Explain why your choice has this property.

Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to have infinitely many solutions. Explain why your choice has this property.