Solving the matrix equation $q(A)=0$, where $q(x)=x^2+x+1$ and $A \in M_{2\times2}(R)$

Question

I have

$$q(x) = x^2 + x + 1$$

and that $q(A) = 0$ where A is a $A \in M_{2\times2}(R)$.

How would I solve this?

This is early on in my course so we haven't been introduced to determinants, eigenvalues or characteristic equations yet. I know how to solve this for complex arguments but struggling on how even to start for real arguments. All I know is that it is not a diagonal matrix (because no real number in $q(x)$ would give $0$).

If you are not familiar with ebdomorphism/matrix reduction, I guess you could also set $$A=\left(\begin{array}{cc}a&b\ c&d\end{array}\right)$$ and solve for $a$, $b$, $c$ and $d$. — Kolakoski54, Jun 29 '24 at 14:41

score 0 · Answer 1 · answered Jun 29 '24 at 15:00

If we multiply $x^2+x+1$ by $x-1$, we get $x^3-1$. So if $A$ is a solution to $x^2+x+1=0$, it is also a solution to $x^3-1=0$. Thus we are looking for a $2\times 2$ non-identity real matrix $A$ such that $A^3=I$.

Can you think of such a matrix? Hint: Think about rotations of the plane...

Of course, after you find a solution to $x^3-1$ by this method, you need to check your solution really is a solution to $x^2+x+1$. This doesn't follow immediately, since the ring $M_{2\times2}(\mathbb{R})$ has zero-divisors.

Solving the matrix equation $q(A)=0$, where $q(x)=x^2+x+1$ and $A \in M_{2\times2}(R)$

1 Answers1