A \(3 \times 3\) real matrix need not have any real eigenvalues.

Question

True or False - and Why?
a) A \(3 \times 3\) real matrix need not have any real eigenvalues.
b) If an \(n \times n\) matrix \(A\) is invertible, then it is diagonalizable.
c) If \(A\) is a \(2 \times 2\) matrix both of whose eigenvalues are 1 , then \(A\) is the identity matrix.
d) If \(\vec{v}\) is an eigenvector of the matrix \(A\), then it is also an eigenvector of the matrix \(B:=A+7 I\).