Chapter 7 - Eigenvalues and Eigenvectors - 7.1 The Eigenvalue/Eigenvector Problem - True-False Review - Page 443: e

True

If $A$ is the $2 × 2$ matrix of the linear transformation $T : R^2 \rightarrow R^2$ that rotates points of the $xy-plane$ counterclockwise by 90 degrees, then $A$ has no real eigenvalues. Because 90 degrees rotation changes the direction of the vector and it makes $v$ and $Av$ not parallel.