Consider the quadratic form

f : R² R

(x₁ x₂) 4(x₁ + 3x₂)² + 3(2x₁ x₂)².

a) Find matrices P, D M₂(R), with D diagonal, such that

f(x) = (P^Tx)^TD(P^Tx)

for all x 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 1y1² + 2y2² for some new variables y₁, y₂. How are the vectors (x₁, x₂) and (y₁, y₂) related?

b) If A M₂(R) is the symmetric matrix associated to f, show that A = P DP T. You may use the fact that if B, C Mn(R) are symmetric and x^TBx = x^TCx for all x Rⁿ, then B = C.