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
$$\mathrm A = \det(\mathrm A) \, 1_2 1_2^{\top} + \begin{bmatrix} 0 & a\\ b & c\end{bmatrix}$$
Assuming that $a b \neq 0$ and using the matrix determinant lemma,
$$\begin{array}{rl} \det(\mathrm A) &= \left( 1 + \det(\mathrm A) 1_2^{\top} \begin{bmatrix} 0 & a\\ b & c\end{bmatrix}^{-1} 1_2 \right) \det \begin{bmatrix} 0 & a\\ b & c\end{bmatrix}\\ &= \left( 1 - \frac{\det(\mathrm A)}{a b} 1_2^{\top} \begin{bmatrix} c & -a\\ -b & 0\end{bmatrix} 1_2 \right) (- a b)\\ &= - a b + \det(\mathrm A) 1_2^{\top} \begin{bmatrix} c & -a\\ -b & 0\end{bmatrix} 1_2\\ &= (c - a - b) \det(\mathrm A) - a b\end{array}$$
