I'm trying to show that any matrix X with rank n can be written as the sum of matrices Z and Y with rank n-1 and 1, respectively. Since X,Y, Z have the same dimensions, is this a simple matter of saying pick one of the columns in X with a pivot. Let Z= X with this column replaced by zeroes but all other entries the same as X. Let Y consist of the removed column and all other entries 0. Thus, X=Y+Z and Y has rank n-1 and Z has rank 1. Does this look correct?