How do i generalize theory to arbitrary trace and arbitrary determinant?

Question

Given a matrix $A\in \Bbb R^{2\times 2}$
Assume that trace $A = 0$. Then:
a.
If $\det A = 0$, then $0$ is the only eigenvalue.
b.
If $\det A <0$, then eigenvalue is $\pm\sqrt{-\det A}$
c.
If $\det A >0$, then eigenvalue is $\pm i \sqrt{\det A}$

I am a bit lost in how do i generalize this theory to arbitrary trace and arbitrary determinant?

For example, how can I find the eigenvalue if $\det A = 0, \operatorname{tr} A > 0$ and if $\det A < 0 , \operatorname{tr} A > 0$

Thanks a lot!

Answer 1

The determinant $d$ is the product of the eigenvalues, which I'll call $a$ and $b$; the trace $t$ is the sum. If you know the det is zero, we have $$ a \cdot b = 0\\ a + b = t $$ from which we conclude that $a = 0$ (we can choose which eigenvalue we want to call zero) and then conclude that $b = t$. You can do similar things for the other possible cases.