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

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