These two questions are the exercises in Hungerford's Algebra(p.462 ex.1 and ex.5).
1. If A is afinite dimensional central simple algebra over the field K, then
A tensor the opposit ring of A over K is isomorphic to a space of n by n
matrix over K. Where n is the dimension of A over K.
2. If A is a finite dimensional central simple algebra over a field K, then
the dimension of A over K is a perfect square.