Find a change of variables that reduces the quadratic form to a sum or differenc

Question

Find a change of variables that reduces the quadratic form to a sum or difference of squares, and express the quadratic form in terms of the new variables. 9x21 + 8x22 + 7x23 - 4x1x2 + 4x2x3 A substitution x = Py that eliminates cross-product terms is x1 = -y1 - 2y2 - 2y3, x2 = - 2y1 - y2 + 2y3, x3 = 2y1 + 2y2 + y3. The new quadratic form is 11y21 + 5y22 + 8y23. A substitution x = Py that eliminates cross-product terms is X1 = -1/3y1 + 2/3y2 - 2/3y3, x2 = -2/3y1 + 1/3y2 + 2/3y3, x3 = 2/3y1 + 2/3y2 + 1/3y3. The new quadratic form is 5y21 + 8y22 + 11y23. A substitution x = Py that eliminates cross-product terms is x1 = -y1 + 2y2 - 2y3, x2 = -2y1 + y2 + 2y3, x3 = 2y1 + 2y2 + y3. The new quadratic form is 5y21 + 8y22 + 11y23. A substitution x = Py that eliminates cross-product terms is x1 = -1/3y1 + 2/3y2 - 2/3y3, x2= -2/3y1 + 1/3y2 + 2/3y3, x3 = -2/3y1 + 2/3y2 - 1/3y3 . The new quadratic form is 5y21 - 8y22 + 11Y23. A substitution x = Py that eliminates cross-product terms is x1 = -1/3y1 - 2/3y2 - 2/3y3, x2 = -2/3y1 - 1/3y2 + 2/3y3, x3 = 2/3y1 + 2/3y2 + 1/3y3. The new quadratic form is 8y21 + 7y22 + 5y23.

Explanation / Answer

4rth option ...A substitution x =Py that eliminates cross product terms is X1 =-1/3 y1 +2/3 y2-2/3y3........

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