I have a quick question about whether or not a matrix is invertible. The question asked is pretty simple, "Suppose that A is a square matrix such that det(A^4) = 0. Show that A cannot be invertible." I know how to explain it, but I'm not sure if it's really the "correct" way, as in I'm not missing anything or making assumptions I wouldn't be allowed to make on an exam. So my stab at it: det(A^4) = det(AAAA) = detA detA detA detA = 0, therefore det A = 0 If A is invertible, there exists a B such that AB = BA = I det(AB) = det(I) detA detB = 1 Therefore, for the matrix to be invertible, detA must be non-zero, which it isn't Like, that seems right to me, but I'm not sure if I have to do any additional work for the part with the inverse to show I understand it.