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3. Consider the quadratic form f : R2 - R 4(x1 + 3x2)2+ 3(21 -22)2. X 2 (a) Find matrices P, DE M2(R), with D diagonal, such that f (x) = (PT x) TD(PTx) for all x E R2. The matrix P need not be orthogonal. Hint: Begin by writing the expression 4(x1 + 3x2)2 + 3(2x1 -x2)2 in the form my + H245 for some new variables y1, y2. How are the vectors (x1, x2) and (y1, y2) related? (b) If A E M2(R) is the symmetric matrix associated to f, show that A = PDPT. You may use the fact that if B, C E Mn (R) are symmetric and x Bx = x Cx for all x E R", then B = C