<s>[INST] Compute the determinant $4\times 4$ Compute the determinant:
$$
       A= \begin{vmatrix}
        1 & 1 & a+1 & b+1 \\
         1 & 0 & a & b \\
         2 & b & a & b \\
        2 & a & a & b \\
        \end{vmatrix}
$$
I got (in the end): $\det A = (-a-b)(b+a) = -ab -a^2 -b^2 -ba $.
 [/INST] You can preform some row operations, they preserve the determinant (but no multiplying a row and $-1$ for every transopsition). This gives me
$$-\begin{vmatrix}
1&0&a&b\\
0&1&1&1\\
0&0&-(a+b)&-2b\\
0&0&-2a&-(a+b)\\
\end{vmatrix}$$
So determinant is $-[(a+b)^2-4ab]$
</s>