If
$$\det \underbrace{\begin{bmatrix} a&b\\ c&d\end{bmatrix}}_{=: A} = -3$$
calculate the determinant
$$\det \underbrace{\begin{bmatrix} 2&-2&0\\ c+1&-1&2a\\ d-2&2&2b\end{bmatrix}}_{=: B}$$
I have no idea how to approach this problem. I can see that the matrix $A$ is contained in $B$ and transposed with a row switch.
$$\begin{bmatrix}2&-2&0\\c+1&-1&2a\\d-2&2&2b\end{bmatrix}^T = \begin{bmatrix}2&c+1&d-2\\-2&-1&2\\0&2a&2b\end{bmatrix}$$
Then after some row switches: $R_2 \leftrightarrow R_3$, then $R_1 \leftrightarrow R_3$
$$\begin{bmatrix}-2&-1&2\\0&2a&2b\\2&c+1&d-2\end{bmatrix}$$
And with row operation $R_3+R_1$..
$$\begin{bmatrix}-2&-1&2\\0&2a&2b\\0&c&d\end{bmatrix}$$
I can calculate the determinant as $(-2) \cdot (-3) \cdot (2) \cdot (-1)^2$ using cofactor expansion and the properties of determinants. The determinant of the transposed matrix is equal to the determinant of the matrix, the row switches are accounted for by the $(-1)^2$, the row multiplied by 2 is accounted for by $(2)$.
So my answer is 12. Please help me if there's a mistake. Thank you!