LetR = Mn(F) the ring consists of all n*n matrices over a finite field F and
E= E11 + E22 + ... + En-1,n-1, where Eii is the elementary matrix(Eij is matrix whose ij th element is 1 and the others are 0). Then the following hold:
1. RE is a maximal left ideal.
2. If A is a rank n-1 matrix in R then A is similar to E.
what is the proof of the above statements?
thank you
if has rank then one column (we may assume it's the last column) is a linear combination of other columns. then doing some column operations, we can make the entries of the last
column all zero. so every matrix of rank is similar to an element of (note that is exactly the set of all elements of whose last columns are zero). so yes, you may assume that