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

---

Definitions:

Flexible Budget

A budget that adjusts or varies with changes in volume or activity levels.

Overhead Costs

Costs that are not directly related to the creation of products or services, including rent, utilities, and salaries for administrative staff.

Production Capacity

The maximum output that a manufacturing or production facility can achieve within a given period under normal working conditions.

Volume Variance

Refers to the difference between the actual volume of output and the budgeted or expected volume, impacting costs and revenues.