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)
• \[P = \left[ \begin{array} { r r r }
• 2 & - 2 & 1 \\
• 2 & 1 & - 2 \\
• 1 & 2 & 2
• \end{array} \right] ; 25 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 } + y _ { 3 } ^ { 2 }\]

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