Let \(A \in M(n, \mathbb{F})\) have an eigenvalue \(\lambda\) with corresponding eigenvector \(v\).

True or False

a) \(-v\) is an eigenvector of \(-A\) with eigenvalue \(-\lambda\).

b) If \(v\) is also an eigenvector of \(B \in M(n, \mathbb{F})\) with eigenvalue \(\mu\), then \(\lambda \mu\) is an eigenvalue of \(A B\).

c) Let \(c \in \mathbb{F}\). Then \((\lambda+c)^{2}\) is an eigenvalue of \(A^{2}+2 c A+c^{2} I\).

d) Let \(\mu\) be an eigenvalue of \(B \in M(n, F)\), Then \(\lambda+\mu\) is an eigenvalue of \(A+B\).

e) Let \(c \in \mathbb{F}\). Then \(c \lambda\) is an eigenvalue of \(c A\).