Let $\rho ∈ Hom_K(V,V)$ such that $\rho = \rho_1\oplus\rho_2$ with respect to the sum $V = U_1\oplus U_2$ where $U_i, i = 1,2$ is invariant under $\rho$. Show that $p_\rho(λ) = p_{\rho_1}(λ)p_{\rho_2}(λ)$ where $p_\rho,p_{\rho_1},p_{\rho_2}$ are the characteristic polynomials.
My attempt:
Let $\{e_1\}^n_{i=1}$ be a basis for $U_1$ and $\{f_1\}^m_{i=1}$ be a basis for $U_2$, so the basis of V is $\{e_1,...,f_m\}$.
Now, $\rho(e_1) = a_{11}e_1+...+a_{1n}e_n, ...,\rho(e_n) = a_{n1}e_1+...+a_{nn}e_n$, $\rho(f_1) = b_{11}f_1+...+b_{1m}f_n, ...,\rho(f_m) = b_{m1}f_1+...+b_{mm}f_m$
Let
$$ X=\begin{bmatrix} a_{11} & ... & a_{n1} \\ . & & .\\ . & & .\\ . & & .\\ a_{1n} & .. & a_{nn} \\ \end{bmatrix} \quad $$ $$ Y=\begin{bmatrix} b_{11} & ... & b_{m1} \\ . & & .\\ . & & .\\ . & & .\\ b_{1m} & .. & a_{mm} \\ \end{bmatrix} \quad $$
So now, I want a matrix of the form $$ \begin{bmatrix} X & 0 \\ 0 & Y \\ \end{bmatrix} \quad $$
So that I can compute the characteristic polynomial and so that will prove the exercise. But I don't know how to get that matrix.