Question

Let for i = 1, 2, 3, p_i(x) be a polynomial of degree 2 in x, p'_i(x) and p''_i(x) be the first and second order derivatives of p_i(x) respectively. Let,

A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`

and B(x) = [A(x)]^T A(x). Then determinant of B(x) ______

Options

• is a polynomial of degree 6 in x.

• is a polynomial of degree 3 in x.

• is a polynomial of degree 2 in x.

• does not depend on x.

Answer 1

Let for i = 1, 2, 3, p_i(x) be a polynomial of degree 2 in x, p'_i(x) and p''_i(x) be the first and second order derivatives of p_i(x) respectively. Let,

A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`

and B(x) = [A(x)]^T A(x). Then determinant of B(x) is a polynomial of degree 6 in x.

Explanation:

Let p₁(x) = a₁x² + b₁x + c₁

P₂(x) = a₂x² + b₂x + c₂ 

and p₃(x) = a₃x² + b₃x + c₃

where a₁, a₂, a₃, b₁, b₂, b₃, c₁, c₂, c₃ are real numbers.

∴ A(x) = `[(a_1x^2 + b_1x + c_1, 2a_1x + b_1, 2a_1),(a_2x^2 + b_2x + c_2, 2a_2x + b_2, 2a_2),(a_3x^2 + b_3x + c_3, 2a_3x + b_3, 2a_3)]`

B(x) = `[(a_1x^2 + b_1x + c_1, a_2x^2 + b_2x + c_2, a_3x^2 + b_3x + c_3),(2a_1x + b_1, 2a_2x + b_2, 2a_3x + b_2),(2a_1, 2a_2, 2a_3)]`

`x[(a_1x^2 + b_1x + c_1, 2a_1x + b_1, 2a_1),(a_2x^2 + b_2x + c_2, 2a_2x + b_2, 2a_2),(a_3x^2 + b_3x + c_3, 2a_3x + b_3, 2a_3)]`

It is clear from the above multiplication, the degree of determinant of B(x) can not be less than 4.