Geometric interpretation of eigenvectors

If $A$ is the matrix of a linear transformation that rotates all vectors in $\mathbb{R}^{2}$ by $60^{\circ}$, explain why $A$ cannot have any real eigenvalues. Is there an angle such that rotation by this angle would have a real eigenvalue? What eigenvalues would be obtainable in this way?

Let $T$ be the linear transformation that reflects vectors about the $x$-axis. Find a matrix for $T$ and then find its eigenvalues and eigenvectors.

Let $T$ be the linear transformation that reflects vectors about the line $x=y$. Find a matrix of $T$ and then find eigenvalues and eigenvectors.

Let $T$ be the linear transformation that reflects all vectors in $\mathbb{R}^{3}$ about the $xy$-plane. Find a matrix for $T$ and then obtain its eigenvalues and eigenvectors.