Orthogonally Diagonalize the Matrix, Giving an Orthogonal Matrix P and a Diagonal

question 7

Multiple Choice

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
- $$\left[ \begin{array} { r r r } 13 & 10 & 4 \\10 & 12 & 6 \\4 & 6 & 5\end{array} \right]$$

• A)
• \(\mathrm { P } = \left[ \begin{array} { r r r } 2 & - 2 & 1 \\ 2 & 1 & - 2 \\ 1 & 2 & 2 \end{array} \right] , \mathrm { D } = \left[ \begin{array} { r r r } 25 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array} \right]\)
• B)
• \[P = \left[ \begin{array} { r r r }
• 2 / 3 & - 2 / 3 & 1 / 3 \\
• 2 / 3 & 1 / 3 & - 2 / 3 \\
• 1 / 3 & 2 / 3 & 2 / 3
• \end{array} \right] , D = \left[ \begin{array} { c c c }
• 1 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 25
• \end{array} \right]\]
• C)
• \[P = \left[ \begin{array} { r r r }
• 2 / 3 & - 2 / 3 & 1 / 3 \\
• 2 / 3 & 1 / 3 & - 2 / 3 \\
• 1 / 3 & 2 / 3 & 2 / 3
• \end{array} \right] , D = \left[ \begin{array} { r r r }
• 25 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 1
• \end{array} \right]\]
• D)
• \[P = \left[ \begin{array} { r r r }
• 2 & - 2 & 1 \\
• 2 & 1 & - 2 \\
• 1 & 2 & 2
• \end{array} \right] , D = \left[ \begin{array} { c c c }
• 1 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 25
• \end{array} \right]\]

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