Let $A$ be a finite dimensional unital algebra over a field $K$. Let $a,b \in A$ such that $ab=1$, show that $ba=1$
One way to prove this is by defining multiplication map and then using Rank nuility theorem. Is there any other way using matrices or something else?
Fix a vector space basis $\{a_1, a_2, \ldots, a_n\}$ of the algebra $\mathfrak A$. Then to every $a\in A$ it corresponds a unique $n\times n$ matrix $A$ by the formula $$ \sum_{ij}(A^i_j x^j )a_i = a(\sum_h x^h a_h).$$ To the product $ab$ there corresponds the matrix product $AB$. So this map is an injective algebra homomorphism, hence it is an isomorphism of $\mathfrak A$ onto an algebra of matrices. The property about left and right inverses in $\mathfrak A$ now follows from the analogous one for matrices.