Perhaps (perhaps) the definition you have covered is: is non singular iff implies .
If so, suppose is non singular then,
which implies is non singular.
Try the second part.
a. Show that if A(nxn) is a nonsingular matrix, so is A^2.
b. Generalize to n: Show that if A is a nonsingular matrix, so is A^n.
I understand that a nonsingular matrix is one that has an inverse (in our course we have not yet talked about determinants so please do not use those in responses), but I cannot think how I would generally any matrix A to always be nonsingular. If I could figure this out, I think I could prove how A^2 or A^n would follow as being nonsingular as well. Any help on this proof you could offer me is greatly appreciated! Thanks!
I'm not sure I'm understanding your definition of nonsingularity. My professor taught us that if you can find an inverse of a matrix, then it is nonsingular. This is not necessarily when talking about systems of equations which is what I'm getting from your definition, talking about x=0. Why would A or x have to equal 0 to be nonsingular?