Determining values of a coefficient for which a system is and isn't consistent. Given the system : \begin{array}{ccccrcc}
x & + & 2y & + & z & = & 3 \\
x & + & 3y & - & z & = & 1 \\
x & + & 2y & + & (a^2-8)z & = & a
\end{array}
Find values of $a$ such that the system has a unique solution, infinitely many solutions, or no solutions.

I begin by placing the system into an augmented matrix.
$\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
1 & 3 & -1 & 1 \\
1 & 2 & a^2-8 & a \\
\end{array}
\right]
$
I then perform operations on the matrix.
$\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
1 & 3 & -1 & 1 \\
1 & 2 & a^2-8 & a \\
\end{array}
\right]
$ $: (R_3-R_1)\rightarrow$ $\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
1 & 3 & -1 & 1 \\
0 & 0 & a^2-9 & a-3 \\
\end{array}
\right]
$
$\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
1 & 3 & -1 & 1 \\
0 & 0 & a^2-9 & a-3 \\
\end{array}
\right]
$ $: (R_2-R_1; R_3/(a-3))\rightarrow$ $\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
0 & 1 & -2 & -2 \\
0 & 0 & a+3 & 1 \\
\end{array}
\right]
$
$\displaystyle
\left[
\begin{array}{rrr|r}
1 & 2 & 1 & 3 \\
0 & 1 & -2 & -2 \\
0 & 0 & a+3 & 1 \\
\end{array}
\right]
$ $: (R_1-2R_2)\rightarrow$ $\displaystyle
\left[
\begin{array}{rrr|r}
1 & 0 & 5 & 3 \\
0 & 1 & -2 & -2 \\
0 & 0 & a+3 & 1 \\
\end{array}
\right]
$

Truth be told, at this point, I'm not too clear on how to progress (or if my approach is even ideal for this issue). I desire to find intervals across all of $a\in\mathbb{R}$ that explain where $a$ causes the system to have a unique, infinite, or inconsistent solution set.

What strikes me immediately
is that if
$a^2-8 = 1$,
the first and last equations
have the same LHS.
Since choosing $a=3$
makes these equations
identical,
this becomes only two equations
in three unknowns,
so there are an
infinite number of solutions.
On the other hand,
if you choose
$a=-3$,
then the first and third equations
are contradictory,
so there are no solutions
in this case.
I will leave the discussion
of other values of $a$ 
to others.
