Consider the system
\begin{eqnarray*}
x + y - z & = & 2 \\
-x - 2y + 3z & = & 1 \\
y + z & = & 0
\end{eqnarray*}
After performing two elementary operations,
one obtains
\begin{eqnarray*}
y + z & = & 0 \\
- y + 2z & = & a \\
x + y - z & = & 2
\end{eqnarray*}
What must be the value of \(a\)?
The answer is \(3\).
We are told that exactly two elementary operations
have been performed. They cannot both be interchanging two equations because
the second equation of the resulting system does not appear in the original
system. Note that one of the operations must be interchanging the first
and third equations.
Doing that gives
\begin{eqnarray*}
y + z & = & 0 \\
-x - 2y + 3z & = & 1 \\
x + y - z & = & 2 \\
\end{eqnarray*}
To bring this system to the final system, we must add the third equation to
the second equation, giving
\begin{eqnarray*}
y + z & = & 0 \\
- y + 2z & = & 3 \\
x + y - z & = & 2
\end{eqnarray*}