Consider the following system:
\begin{eqnarray}
y - 2z + x & = 4\\
y +2z & = 5
\end{eqnarray}
Here, \(x, y,\) and \(z\) are the variables.
The system is equivalent to the following system in matrix form:
\[
\begin{bmatrix} 1 & 1 & a \\ 0 & 1 & 2 \end{bmatrix}
\begin{bmatrix} x \\ y \\ z \end{bmatrix}
=
\begin{bmatrix} 4 \\ 5 \end{bmatrix}.
\]
What must be the value of \(a\)?
The answer is \(-2\).
This is a bit of a trick question as the variables do not appear in the
order one expects them to appear. The best way to do this is to write
out the matrix form in full:
\begin{eqnarray}
x + y + az & = 4\\
y -2z & = 5
\end{eqnarray}
Comparing the first equation of this system with the original,
one sees that the coefficient of \(z\) is \(-2\). So \(a = -2\).