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!