Question

Let \(a, b, c, d\), and \(e\) be real numbers. For each of the following matrices, find their eigenvalues, corresponding eigenvectors, and orthogonal matrices that diagonalize them.
\[
A=\left(\begin{array}{ll}
a & b \\
b & a
\end{array}\right), \quad B=\left(\begin{array}{lll}
a & b & 0 \\
b & a & 0 \\
0 & 0 & c
\end{array}\right), \quad C=\left(\begin{array}{lllll}
a & b & 0 & 0 & 0 \\
b & a & 0 & 0 & 0 \\
0 & 0 & c & d & 0 \\
0 & 0 & d & c & 0 \\
0 & 0 & 0 & 0 & e
\end{array}\right) .
\]