Statements about complex eigenvalues

Question

Suppose that $A$ is a $3 \times 3$ real matrix with $k$ distinct real eigenvalues and $l$ distinct complex (i.e., non-real) eigenvalues. Consider the following statements:

$(i)$ $k=2,l=0$

$(ii)$ $k=0,l=3$

$(iii)$ $k=1,l=2$

$(iv)$ $k=2,l=1$

How many of the above statements are possible?

$(A)$ None

$(B)$ $1$

$(C)$ $2$

$(D)$ $3$

$(E)$ all of them

I think that only statements $(i)$ and $(iii)$ are possible. Complex numbers come in conjugate pairs so $l \neq 3,1$ So that means either $l=0,2$. Thus, only statements $(i)$ and $(iii)$ are possible so the answer is $(C)$.

Is that the correct train of thought? Thanks!

Answer 1

You also have to demonstrate that $(i)$ and $(iii)$ are indeed possible. Giving an example of each case is probably the easiest way to do that. Might I suggest $$ \begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix} $$ for $(i)$ and $$ \begin{bmatrix}1&0&0\\0&0&-1\\0&1&0\end{bmatrix} $$ for $(iii)$?

Also, "Complex numbers come in conjugate pairs" is possibly a bit vague. A better way of saying that would be something like "The characteristic equation is a polynomial equation with real coefficients, and complex solutions to such equations come in conjugate pairs."

Answer 2

For this its better we consider the characteristic equation of $A$.

$(i)$ Characteristic equation = $(\lambda-a)^2(\lambda-b) \qquad a,b\in\mathbb{R}\qquad$ and

$(iii)$ Characteristic equation = $(\lambda-a)(\lambda^2 +b)\qquad a\in\mathbb{R},\;b\in\mathbb{R}^+\qquad$ are possible.

For $(ii)$ and $(iv)$, note that complex eigenvalues, concerning the characteristic equation with real coefficients, always occur in conjugate pairs.

@FutureMathPerson: Your arguments for $l$ are correct but your answer is incomplete since you have not mentioned anything about $k$.