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

question 25

Multiple Choice

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
- $$\left[ \begin{array} { r r r } 11 & 7 & 7 \\7 & 11 & 7 \\7 & 7 & 11\end{array} \right]$$

• A)
• \[P = \left[ \begin{array} { c c c }
• 1 / \sqrt { 3 } & - 1 / \sqrt { 2 } & - 1 / \sqrt { 6 } \\
• 1 / \sqrt { 3 } & 1 / \sqrt { 2 } & - 1 / \sqrt { 6 } \\
• 1 / \sqrt { 3 } & 0 & 2 / \sqrt { 6 }
• \end{array} \right] , D = \left[ \begin{array} { r r r }
• 4 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 25
• \end{array} \right]\]
• B)
• \[P = \left[ \begin{array} { r r r }
• 1 & - 1 & - 1 \\
• 1 & 1 & - 1 \\
• 1 & 0 & 2
• \end{array} \right] , D = \left[ \begin{array} { r r r }
• 25 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 4
• \end{array} \right]\]
• C)
• \[P = \left[ \begin{array} { r c c }
• 1 / \sqrt { 3 } & 1 / \sqrt { 2 } & 1 / \sqrt { 6 } \\
• 1 / \sqrt { 3 } & - 1 / \sqrt { 2 } & 1 / \sqrt { 6 } \\
• - 1 / \sqrt { 3 } & 0 & 2 / \sqrt { 6 }
• \end{array} \right] , D = \left[ \begin{array} { r r r }
• 25 & 0 & 0 \\
• 0 & 25 & 0 \\
• 0 & 0 & 4
• \end{array} \right]\]
• D)
• \[P = \left[ \begin{array} { c c c }
• 1 / \sqrt { 3 } & - 1 / \sqrt { 2 } & - 1 / \sqrt { 6 } \\
• 1 / \sqrt { 3 } & 1 / \sqrt { 2 } & - 1 / \sqrt { 6 } \\
• 1 / \sqrt { 3 } & 0 & 2 / \sqrt { 6 }
• \end{array} \right] , D = \left[ \begin{array} { r r r }
• 25 & 0 & 0 \\
• 0 & 4 & 0 \\
• 0 & 0 & 4
• \end{array} \right]\]

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