If $X$ is a finite dimensional irreducible representation of the matrix algebra of order $m$ over $A$, where $A$ is a finite dimensional algebra over a field $F$, is it true that $X$ is isomorphic to the direct product of $m$ irreducible representations of $A$?
I think I can see why the RHS is also a simple representation of the matrix algebra. My attempt is to consider $m=2$ first and a nonzero element $x\in X$, then construct the set of matrices with all but one certain column is zero acting on $x$. However I am still confused and cannot find the desired irreducible representations. Maybe this is a wrong approach.
Any guidance is really appreciated. Thank you!