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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 $A=PDP_{T}$ 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 $u$ and $v$ satisfy $Au=4u$ and $Av=3v$, then $u?v=0$.
The eigenvalues and corresponding eigenvectors of a symmetric matrix A are the following: $?_{1}=?4,?_{2}=2,?_{3}=?1,u_{1}?=???210????,u_{2}?=???1?21????,u_{3}?=????125????$, Find matrices D and $P$ for an orthogonal diagonalization for A. Use the order in which the eigenvalues are given.