Chapter 7.2, Problem 46E

In all the parts of this problem, consider a matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ with the eigenvalues ${\lambda }_1$ and ${\lambda }_2$ .
a. Show that ${\lambda }_1^2+{\lambda }_2^2=a^2+d^2+2bc$ .
b. Show that ${\lambda }_1^2+{\lambda }_2^2\le a^2+d^2+c^2+d^2$ .
c. For which matrices A does the equality ${\lambda }_1^2+{\lambda }_2^2=a^2+d^2+c^2+d^2$ hold?