(Solved): True or False: select all true statements. A. If a symmetric matrix A has eigenvalues 2,1,0, ...

True or False: select all true statements. A. If a symmetric matrix $A$ has eigenvalues $?2,?1,0$, then $A$ cannot be written as $B^{T}B$ for any matrix $B$. B. If $A=PD{P}^T$ where $P^T=P^{?1}$ and $D$ is a diagonal matrix, then $A$ is symmetric. C. There are symmetric matrices that are not orthogonally diagonalizable. D. If $A^{T}A=I$, then $A$ is an orthogonal matrix. E. If $A^T=A$ and if vectors $\vec{u}$ and $\vec{v}$ satisfy $A\vec{u}=4\vec{u}$ and $A\vec{v}=3\vec{v}$, then $\vec{u}?\vec{v}=0$. The eigenvalues and corresponding eigenvectors of a symmetric matrix A are the following: ${?}_1=?4,{?}_2=2,{?}_3=?1,\overrightarrow{u_1}=?\begin{array}{c}2\\ 1\\ 0\end{array}?,\overrightarrow{u_2}=?\begin{array}{c}1\\ ?2\\ 1\end{array}?,\overrightarrow{u_3}=?\begin{array}{c}?1\\ 2\\ 5\end{array}?$, Find matrices D and $\mathrm{P}$ for an orthogonal diagonalization for A. Use the order in which the eigenvalues are given.