<s>[INST] Find all values of a for which the following system has a solution, no solution and infinitely many solution. Find all values of a for which the following system
$$\begin{cases}x+2y+z=a^2 
\\x+y+3z=a
\\3x+4y+8z=8\end{cases}$$
has a solution, no solution and infinitely many solution.
I found the reduced row echelon form of this system which is:
$\begin{pmatrix}
1 & 0 & 0 & 4a^2+12a-40\\
0 & 1 & 0 & -a^2-5a+16\\
0 & 0 & 1 & -a^2-2a+8
\end{pmatrix}$
Does that mean this system has only one solution and there is no value for a which makes the system infinitely many solutions and no solution?
 [/INST] Yes since the RREF is the following
$$\begin{pmatrix}
1 & 2 & 1 &a^2\\
1 & 1 & 3 & a\\
3 & 4 & 8 & 8
\end{pmatrix} \to \begin{pmatrix}
1 & 2 & 1 &a^2\\
0 & 1 & -2 & a^2-a\\
0 & 0 & 1 & 8-2a-a^2
\end{pmatrix}$$
by Rouché–Capelli theorem, we always have exacty one soution for the system.
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