Find all values of h such that rank(A) = 2

Question

Find all values of $h$ such that rank($A$) = $2$.

$A$ = $\begin{bmatrix}
1 & h & -1\\
3 & -1 & 0\\
-4 & 1 & 3
\end{bmatrix} $

I used row transformations to get

$A$ = $\begin{bmatrix}
1 & h & -1\\
0 & -1-3h & 3\\
0 & 1+4h & -1
\end{bmatrix} $

But how do I solve to get the rank? I know the general idea is that rank($A$) = $2$ when dim(col($A$)) = dim(row($A$)) = $2$

Answer 1

You know that the first row is independent, but the second and third rows must be dependent. In other words, you can write $$-1-3h=C(1+4h)\\3=C(-1)$$ Therefore $C=-3$. Plug it into the first equation, to find $h$