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 firs…

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 $A(x)=\begin{pmatrix}p_1 & p_1' & p_1''\p_2 & p_2' & p_2''\p_3 & p_3' & p_3''\end{pmatrix}$ and $B(x)=[A(x)]^TA(x)$. Then determinant of $B(x)$:

Accepted Answer

$\det B(x)=\det(A^T)\det(A)=[\det A(x)]^2$. Since each entry of $A(x)$ is at most quadratic in $x$, $\det A(x)$ is a polynomial of degree at most 3 in $x$ (generically degree 3, from a constant multiple of the Vandermonde-type structure of quadratics and their derivatives). So $\det B(x)=[\det A(x)]^2$ has degree $6$.

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