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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 $c A$.
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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 $c A$.
by Platinum (93,241 points)

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