Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

Question

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix}

Find the values of a and b such that the system has:

a) A unique solution

So I think I got this although I'm not sure.

$|A| \not= 0$

reduced it down to:

\begin{bmatrix} 1 & 0 & b/a \\ 0 & 1 & 4-1/a \\ 0 & 0 & b-2 \\ \end{bmatrix}

Therefore $b\not=2$ and $a\not=0$ Do I mention anything else here?

b) A two-parameter solution

This is the one I'm having trouble with. I don't know how to answer it and can't find anything telling me how to.

I got this from simplifying everything down and assuming $y=v$ and $z=w$

$$x=\frac{4-av-4w}{a}$$

Therefore $a\not=0$ and $b$=all real

Answer 1

Here is one way to approach this problem:

Reduce $\begin{bmatrix}a&0&b&2\\a&a&4&4\\0&a&2&b\end{bmatrix}\longrightarrow\begin{bmatrix}a&0&b&2\\0&a&4-b&2\\0&a&2&b\end{bmatrix}\longrightarrow\begin{bmatrix}a&0&b&2\\0&a&4-b&2\\0&0&b-2&b-2\end{bmatrix}$.

Now continue reduction to row echelon form in each of the following cases:

1) $a\ne0, b\ne 2$

2) $a\ne0, b=2$

3) $a=0, b=2$

4) $a=0, b\ne2$