Rank of a matrix with two parameters

Question

Find the rank of the matrix $A$:

$A = \begin{bmatrix} a & 1-a & 1 \\ b & -a & b \\ a-b & b & 1\end{bmatrix}$ with $a , b \in \mathbb{R}$

I know that I could take the determinant to check for rank 3, but how would I do it for cases of rank 2 and 1?

Answer 1

Hint: compute a row-echelon form of $A$. The rank is the number of non-zero lines in such a form.