For a, b ∈ \(\mathbb{R}\), let

p(x, y) = a²x₁ y₁ + abx₂y₁ + abx₁y₂ + b²x₂y₂, x =(x₁, x₂), y = (y₁, y₂) ∈ \(\mathbb{R}\)²

For what values of a and b does p ∶ \(\mathbb{R}\)² × \(\mathbb{R}\)² → \(\mathbb{R}\) define an inner product? 

Explanation:

∵ p(x, y) is bilinear transform. then matrix representation of p(x, y) is, 

\(A=\left[\begin{array}{ll} a^2 & a b \\ a b & b^2 \end{array}\right]\)

If p(x, y) is defining an inner product if A is positive definite i.e. all eigenvalues of A are positive.

but det (A) = 0 ⇒ all eigenvalues can't be positive.

(∵ at least one eigenvalue of A

⇒ A is not positive definite.

⇒ p(x, y) is not defining an inner product for any value of a, b.

 Option (4) is true