Homomorphism Between 2x2 matrices and the Reals

Question

Let $A=\begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$.

1. Let $\mathfrak{A}_\mathbb{R}$ be the real unital subalgebra of $M_2(\mathbb{R})$ generated by the matrix $A$. Prove that there are no nonzero homomorphisms $\varphi:\mathfrak{A}_\mathbb{R} \rightarrow \mathbb{R}$.
2. Let $\mathfrak{A}_\mathbb{C}$ be the complex unital subalgebra of $M_2(\mathbb{C})$ generated by $A$. Prove that there is at least one nonzero homomorphism $\rho:\mathfrak{A}_\mathbb{C} \rightarrow \mathbb{C}$.

I'd appreciate any suggestions/hints/insights!

Answer 1

Hint: $A^2=-I$ and so $A$ behaves like $i \in \mathbb C$.

Solution for 1:

$\varphi(A)^2=\varphi(A^2)=\varphi(-I)=-\varphi(I)=-1$ cannot happen in $\mathbb R$.

Solution for 2:

The elements of $\mathfrak{A}_\mathbb{C}$ are of the form $aI+bA$ with $a,b \in \mathbb C$. Define $\varphi(aI+bA)=a+bi$.