(Solved): Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues ...

Matrix A is factored in the form PDP ${}^{?1}$. Use the Diagonalization Theorem to find the eigenvalues of $\mathrm{A}$ and a basis for each eigenspace. $A=?\begin{array}{lll}2& 1& 2\\ 1& 2& 2\\ 1& 1& 3\end{array}?=?\begin{array}{rrr}2& 2& 1\\ 2& 0& ?1\\ 2& ?1& 0\end{array}??\begin{array}{lll}5& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}??\begin{array}{ccc}\frac{1}{8}& \frac{1}{8}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& ?\frac{1}{2}\\ \frac{1}{4}& ?\frac{3}{4}& \frac{1}{2}\end{array}?$ Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A. There is one distinct eigenvalue, $?=$. A basis for the corresponding eigenspace is B. In ascending order, the two distinct eigenvalues are ${?}_1=$ and ${?}_2=$ Bases for the corresponding eigenspaces are ' and ', respectively. C. In ascending order, the three distinct eigenvalues are ${?}_1=,{?}_2=$, and ${?}_3=$. Bases for the corresponding eigenspaces are $?,?$, and, , respectively.