Characteristic Roots of Matrix

[Guest]

Can someone solve this, or at least gave me a hint?
a) Prove that a square matrix A has non-zero determinant (i.e. non-singular) if, and only if, the characteristic equation of A has all non-zero roots.
b) Show that if a 4x4 matrix has determinant zero, then one of the root of characteristic equation is zero.

[Guest]

Or- For any matrix, A, there exist an invertible matrix, B, such that BAB^-1= c, the "Jordan Normal Form" of A, a upper triangular matrix having the eigenvalues of A on the diagonal. So det(C)= det(B)det(A)det(B^-1)= det(A)det(B)det(B^-1)= det(A). The determinant of any triangular matrix is the product of the numbers on the diagonal so det(A) is non-zero if and only if C has no zeros on its diagonal= A has no zero eigenvalues.