Let A be an n ( n matrix.
(a) Show that det(A) is the product of all the roots of the characteristic polynomial of A.
(b) Show that A is singular if and only if 0 is an eigenvalue of A.
(c) Also prove the analogous statement for a linear transformation: If L: V → V is a linear transformation, show that L is not one-to-one if and only if 0 is an eigenvalue of L.
(d) Show that if A is nilpotent then A is singular?