I have the following matrix:
$$\begin{pmatrix} b+3 & a & 4 & -2b-1\\ b & -3 & 5 & -6\\ -1 & 1 & 2a+1 & 1-a \end{pmatrix}$$
How can I determine the rank for different values of $a$ and $b$? My initial approach was to simplify the matrix by performing elementary row/column operations. In particular, multiplying the last column with two and adding the third to it:
$$\begin{pmatrix} b+3 & a & 4 & -4b+2\\ b & -3 & 5 & -7\\ -1 & 1 & 2a+1 & 3 \end{pmatrix}$$
From there, I could conclude that the rank is at least 2, since it has a non-zero minor $M_{2,3}^{2,4}$: $$\begin{vmatrix} -3 & -7 \\ 1 & 3 \end{vmatrix} = -2$$
Expanding the minor by one row and column yields two $3 \times 3$ minors, but neither of them gets me a reasonably simple expression. I also tried to transform the matrix into row echelon form, but the values get complicated, which makes it hard to check the special cases one by one. Any ideas?