1. singular matrices, transposes etc

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

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

manny thanks

2. is the transpose of a singular Matrix singular?
The answer is yes. A matrix $A$ is singular if and only if its determinant is equal to 0.
There's a proof that says that $\det A= \det A^T$. So if $A$ is singular, so is its transpose. And if a $A^T$ is singular, so is $A$.

what does row reducing (matrix|identity) do??
You cannot form "identity" since you don't have a square matrix.

3. but what about if you have an n x m matrix and augment it with an m x m identity, then row reduce? what happens?

(thanks)

4. Originally Posted by James0502
but what about if you have an n x m matrix and augment it with an m x m identity, then row reduce? what happens?

(thanks)
Nothing would happen. If you do that and say that m>n, then you wouldn't be able to touch the lowest rows of the identity matrix.
You can only do that when you have a square matrix. And this method finds the inverse of the matrix you chose, if it has an inverse. If you chose a non square matrix it's not even worth to bother trying to reduce it with an mxm identity matrix aside it. As I said, you wouldn't find anything.