On the space of $2\times 2$ matrices, let $T$ be the transformation that transposes every matrix. Find the eigenvalues and "eigenmatrices" for $A^T =\lambda A$.
By taking determinants on the left and right hand sides in the last equation we may show that $\lambda=\pm1$. But I have no idea beyond that.
The transformation is a linear transformation all right but what will be its matrix? What is meant here by "eigenmatrices"?
In the context of this question, an "eigenmatrix" of the transformation $T$ is a matrix $A \in M(2,\Bbb R)$ such that $A^T = \lambda A$ for some scalar $\lambda$.
So, for example, the matrix $$ A = \pmatrix{1&0\\0&0} $$ is an eigenmatrix because $$ T(A) = A^T = A = 1 \cdot A $$