# How to proof that the irreducible representations of the upper triangular $n$ by $n$ matrices are $V_1,…,V_n$?

Here the ground field $k$ is algebriacally closed.$A$ is the algebra of upper triangular $n$ by $n$ matrices.

I already know that $V_i$ which is 1-dimensional, and any matrix $x$ acts by $x_{ii}$($x_{ii}$ is the ($i$,$i$)-th entry of $x$) is irreducible representation of $A$($i=1,2,...,n$).

I know a method to show that these are all the irreducible representations of $A$. It uses that fact that $A/Rad(A)=\bigoplus End(W_i)$ where $W_i$ are the pair-wise nonisomorphic irreducible representations of $A$.

Are there any other more direct method to do this?

Would someone be kind enough to give me some hints on this?Thank you very much!

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