Can you help me prove this statements.
Let A be a ring, and A, B be 2 matrixs. if
$\displaystyle $A_{mxn}B_{nxm}=I_m$$ and $\displaystyle $B_{nxm}A_{mxn}=I_n$$ then m=n.
Im and In are identity marix.
Thank you .
Can you help me prove this statements.
Let A be a ring, and A, B be 2 matrixs. if
$\displaystyle $A_{mxn}B_{nxm}=I_m$$ and $\displaystyle $B_{nxm}A_{mxn}=I_n$$ then m=n.
Im and In are identity marix.
Thank you .
Think about it like this. You have shown that $\displaystyle A$ is an isomorphism of $\displaystyle R^n\to R^m$. Assuming that $\displaystyle R$ is commutative (so that it has the invariant basis number property) then $\displaystyle n=m$.
Best,
Alex