(Solved): Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are o ...

Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. \[ \left[\begin{array}{rrr} -1 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 3 \end{array}\right] \] Find the characteristic polynomial of \( A \). \[ |\lambda I-A|= \] Find the eigenvalues of \( A \). (Enter your answers from smallest to largest.) \[ \left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)=( \] Find the general form for every eigenvector corresponding to \( \lambda_{1} \). (Use \( s \) as your parameter.) \[ \mathbf{x}_{1}= \] Find the general form for every eigenvector corresponding to \( \lambda_{2} \). (Use \( t \) as your parameter.) \[ \mathbf{x}_{2}= \] Find the general form for every eigenvector corresponding to \( \lambda_{3} \). (Use \( u \) as your parameter.) \[ \mathbf{x}_{3}= \] Find \( \mathbf{x}_{1} \cdot \mathbf{x}_{2} \). \[ \mathbf{x}_{1} \cdot \mathbf{x}_{2}= \] Find \( \mathbf{x}_{1} \cdot \mathbf{x}_{3} \). \[ \mathbf{x}_{1} \cdot \mathbf{x}_{3}= \] Find \( \mathbf{x}_{2} \cdot \mathbf{x}_{2} \). \[ \mathbf{x}_{2} \cdot \mathbf{x}_{3}= \] Determine whether the eigenvectors corresponding to distinct eigenvalues are orthogonal. (Select all that apply.) \( \mathbf{x}_{1} \) and \( \mathbf{x}_{2} \) are orthogonal. \( \mathbf{x}_{1} \) and \( \mathbf{x}_{3} \) are orthogonal. \( \mathbf{x}_{2} \) and \( \mathbf{x}_{3} \) are orthogonal.