Suppose that $A$ is a square matrix. Prove that the constant term of the characteristic polynomial of $A$ is equal to the determinant of $A$.

Suppose that $A$ is a square matrix. Prove that a single vector may not be an eigenvector of $A$ for two different eigenvalues.

Suppose that $U$ is a unitary matrix with eigenvalue $\lambda$. Prove that $\lambda$ has modulus 1, i.e. $\modulus{\lambda}=1$. This says that all of the eigenvalues of a unitary matrix lie on the unit circle of the complex plane.

Theorem DCP tells us that the characteristic polynomial of a square matrix of size $n$ has degree $n$. By suitably augmenting the proof of Theorem DCP prove that the coefficient of $x^n$ in the characteristic polynomial is $(-1)^n$.

Theorem EIM says that if $\lambda$ is an eigenvalue of the nonsingular matrix $A$, then $\frac{1}{\lambda}$ is an eigenvalue of $\inverse{A}$. Write an alternate proof of this theorem using the characteristic polynomial and without making reference to an eigenvector of $A$ for $\lambda$.