Determine the $\det(A)$ if $A$ is a eigenvector of a linear operator

Question

Let $T$ be a linear operator defined by $T\left(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = \begin{pmatrix} c & a \\ d & b \end{pmatrix}$ for every $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{R}^{2 \times 2}$. If $A$ is a eigenvector $T$ corresponding to eigenvalues $-1$, then compute $\det(A)$

As long as I have studied about eigenvalues and eigenvectors, I never encounter a problem which take a note about linear transformation. I have a difficulty to find what is the exact component of matrix $A$ in which be used to determine its determinant. Do you have any idea?

Answer 1

HINT: Write down what it means for $T(A) = -A$ to hold, entry by entry.

Alternatively, compare $\det(A)$ and $\det(T(A))$.