Make a Change of Variable, X = Py, That Transforms $$Q ( x ) = 13 x _ { 1 } ^ { 2 } + 12 x _ { 2 } ^ { 2 } + 5 x _ { 3 } ^ { 2 } + 20 x _ { 1 } x _ { 2 } + 8 x _ { 1 } x _ { 3 } + 12 x _ { 2 } x _ { 3 }$$

question 5

Multiple Choice

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product
term. Give P and the new quadratic form.
- $$Q ( x ) = 13 x _ { 1 } ^ { 2 } + 12 x _ { 2 } ^ { 2 } + 5 x _ { 3 } ^ { 2 } + 20 x _ { 1 } x _ { 2 } + 8 x _ { 1 } x _ { 3 } + 12 x _ { 2 } x _ { 3 }$$

• A) \(\mathrm { P } = \left[ \begin{array} { r r r } - 2 / 3 & - 2 / 3 & 1 / 3 \\ - 2 / 3 & 1 / 3 & - 2 / 3 \\ 1 / 3 & - 2 / 3 & 2 / 3 \end{array} \right] ; 5 \mathrm { y } _ { 1 } ^ { 2 } + 2 \mathrm { y } _ { 2 } ^ { 2 } + \mathrm { y } _ { 3 } ^ { 2 }\)
• B)
• \[\mathrm { P } = \left[ \begin{array} { r r r }
• 2 / 3 & - 2 / 3 & 1 / 3 \\
• 2 / 3 & 1 / 3 & - 2 / 3 \\
• 1 / 3 & 2 / 3 & 2 / 3
• \end{array} \right] ; 25 \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 } + \mathrm { y } _ { 3 } ^ { 2 }\]
• C)
• \[\mathrm { P } = \left[ \begin{array} { r r r }
• 2 & - 2 & 1 \\
• 2 & 1 & - 2 \\
• 1 & 2 & 2
• \end{array} \right] ; \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 } + 25 \mathrm { y } _ { 3 } ^ { 2 }\]
• D)
• \[P = \left[ \begin{array} { r r r }
• 2 & - 2 & 1 \\
• 2 & 1 & - 2 \\
• 1 & 2 & 2
• \end{array} \right] ; 25 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 } + y _ { 3 } ^ { 2 }\]

---

Definitions:

Behavioral Economics

An area of economics that incorporates insights from psychology to understand decision-making and economic behavior that deviates from classical economic theories.

Job Offer

A proposal made by an employer to a potential employee detailing the terms of employment, including position, salary, and benefits.

Current Positions

Refers to the existing financial holdings, assets, or job titles held by an individual or entity at a specific point in time.

Organ Donation Programs

Initiatives designed to encourage and facilitate the donation of organs for transplantation from one person to another.