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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.
$A=\left(\begin{array}{ll} a & b \\ b & a \end{array}\right), \quad B=\left(\begin{array}{lll} a & b & 0 \\ b & a & 0 \\ 0 & 0 & c \end{array}\right), \quad C=\left(\begin{array}{lllll} a & b & 0 & 0 & 0 \\ b & a & 0 & 0 & 0 \\ 0 & 0 & c & d & 0 \\ 0 & 0 & d & c & 0 \\ 0 & 0 & 0 & 0 & e \end{array}\right) .$
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