is the transpose of a singular Matrix singular?

and what about if A is singular, is A^T . A singular?

also

I know if you augment a non-singular square matrix with the identity matrix of the same dimension the you will get (inverse|identity) if you reduce it to echelon form

but what about non-square matrices

what does row reducing (matrix|identity) do??? and what about then doing (transpose(matrix)|identity)??

manny thanks