91Ó°ÊÓ

(i) Show that \(\lambda=0\) is an eigenvalue for the \(n \times n\) matrix \(A\) if and only if \(A\) is singular. [Hint: use the \(A-\lambda I\) definition.] (ii) Show that if \(\operatorname{Det}(A-\lambda I) \neq 0\) then \(\lambda\) is definitely not an eigenvalue of \(A\).

Show that if \(A\) has \(\lambda\) as an eigenvalue then \(A+\mu I\) has \(\lambda+\mu\) as an eigenvalue.

Show that the eigenvalues of a skew symmetric matrix are pure imaginary. (That is, are of the form \(a+i b\) with \(a=0\).)