Question

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 \(c A\).

Answer 1

a) True. If \(A v=\lambda v\) then \(-A v=-\lambda v\) and so \(-v\) is an eigenvector of \(-A\) with eigenvalue \(-\lambda\)

b) False. The product of two matrices may not have any eigenvalue equal to the product of the corresponding eigenvalues of the matrices.

c) True. If \(A v=\lambda v\), then \[(A^2 + 2cA + c^2I)v \]

\[= A(Av) + 2cAv + c^2v \]

\[= A(\lambda v) + 2c\lambda v + c^2v \]

\[= (\lambda^2 + 2c\lambda + c^2)v\]

Thus, \((\Lambda+c)^2\) is an eigenvalue of \(A^2+2 c A+c^2 I\).

d) False. The sum of two matrices may not have any eigenvalue equal to the sum of the corresponding eigenvalues of the matrices.

e) True. If \(A v=\lambda v\), then
\[cAv = c \lambda v\].
Thus, \(c \lambda\) is an eigenvalue of \(c A\).