Question

if A is an nxn matrix then if A is diagonalizable is it also invertible

Answer 1

Yes. Since a diagonal matrix can be inverted.

Answer 2

So if the elements in the diagonal of matrix D are d1, d2, ..., dn ; then you can write D = d1*d2*...*dn * I. Now you also have A = PDP^-1 thus, A-1 is just 1/(d1*d2*...*dn) I

Answer 3

No, diagonalizibility does not imply invertibility nor vice versa.

Any diagonalizable matrix with an eigenvalue of zero is not invertible (eg. the matrix used at the beginning of lecture 23). KashyapRaval's answer fails here when one of the diagonal elements is zero since you'll divide by zero. Invertible matrices whose eigenvectors don't form a basis for R^n are not diagonalizable (eg. [0,-1;1,2] is invertible but not diagonalizible).