Compute the determinant $4\times 4$

Question

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 $.

Answer 1

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 $-[(a+b)^2-4ab]$