Equation with matrix and determinant Given
$$A={\begin{bmatrix}
\det(A)  &  \det(A)+a \\
\det(A) +b & \det(A)+c
\end{bmatrix}}$$
where $a,b,c$ are given and $A$ is unknown, is it possible to use some clever tricks concerning determinants for this case? (instead of direct calculations).

Let $\text{det}A=d$. Then we are given
$$A={\begin{bmatrix}
d  &  d+a \\
d +b & d+c
\end{bmatrix}}$$
Using row operations, we get
$$
d=|A|=\begin{vmatrix}
d  &  d+a \\
d +b & d+c
\end{vmatrix} \rightarrow \begin{vmatrix}
d  &  d+a \\
b & c-a
\end{vmatrix}
\rightarrow \begin{vmatrix}
d  &  a \\
b & c-a-b
\end{vmatrix}
$$
Thus 
$$d=d(c-a-b)-ab.$$
Now solve for $d$.
