Show that if A is similar to B, then a. det(A) = det(B). b. The characteristic polynomial

Question:

Show that if A is similar to B, then
a. det(A) = det(B).
b. The characteristic polynomial of A is the same as the characteristic polynomial of B.
c. A is nonsingular if and only if B is nonsingular.
d. If A is nonsingular, show that A−1 is similar to B−1.
e. At is similar to Bt.