Question: For each of the following quadratic forms in three variables, write it in the format \(\mathrm{x}^{t} A \mathrm{x}\), find a substitution \(\mathrm{x}=\mathrm{Py}\) so that it can be written in diagonal form (i.e. as a sum of squares) in terms of the variables \(y_{1}\) \(y_{2}\), and \(y_{3}\). Determine whether the quadratic form is positive definite.

1. \(2 x_{1}^{2}+2 x_{2}^{2}+2 x_{3}^{2}-2 x_{1} x_{2}+4 x_{1} x_{3}+4 x_{2} x_{3}\)
2. \(2 x_{1}^{2}+2 x_{2}^{2}+4 x_{3}^{2}+2 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{2} x_{3}\)
3. \(5 x_{1}^{2}+2 x_{2}^{2}+5 x_{3}^{2}+4 x_{1} x_{2}-8 x_{1} x_{3}-4 x_{2} x_{3}\)

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