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{aligned}\left[\begin{array}{lll} p_1(x) & p'_1(x) & p''_1(x) \ p_2(x) & p'_2(x) & p''_2(x) \ p_3(x) & p'_3(x) &…

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, $$ A(x)=$\begin{aligned}\left[\begin{array}{lll} p_1(x) & p'_1(x) & p''_1(x) \ p_2(x) & p'_2(x) & p''_2(x) \ p_3(x) & p'_3(x) & p''_3(x) \end{array}ight]\end{aligned}$ $$ and $B(x)=[A(x)]^{T} A(x)$. Then the determinant of $B(x)$ is:

Aicharya · Accepted Answer

Let $p_{1}x = a_{1}x^{2} + b_{1}x + c_{1}$, $p_{2}x = a_{2}x^{2} + b_{2}x + c_{2}$, and $p_{3}x = a_{3}x^{2} + b_{3}x + c_{3}$, where $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, c_{1}, c_{2}, c_{3}$ are real numbers. $\begin{aligned} 	herefore A(x) = \left[\begin{array}{ccc} a_{1}x^{2} + b_{1}x + c_{1} & 2a_{1}x + b_{1} & 2a_{1} \ a_{2}x^{2} + b_{2}x + c_{2} & 2a_{2}x + b_{2} & 2a_{2} \ a_{3}x^{2} + b_{3}x + c_{3} & 2a_{3}x + b_{3} & 2a_{3} \end{array}ight] \ = \left[\begin{array}{ccc} a_{1}x^{2} + b_{1}x + c_{1} & a_{2}x^{2} + b_{2}x + c_{2} & a_{3}x^{2} + b_{3}x + c_{3} \ 2a_{1}x + b_{1} & 2a_{2}x + b_{2} & 2a_{3}x + b_{3} \ 2a_{1} & 2a_{2} & 2a_{3} \end{array}ight] \ 	imes \left[\begin{array}{ccc} a_{1}x^{2} + b_{1}x + c_{1} & 2a_{1}x + b_{1} & 2a_{1} \ a_{2}x^{2} + b_{2}x + c_{2} & 2a_{2}x + b_{2} & 2a_{2} \ a_{3}x^{2} + b_{3}x + c_{3} & 2a_{3}x + b_{3} & 2a_{3} \end{array}ight] \end{aligned}$ It is clear from the above multiplication, the degree of determinant of $B(x)$ cannot be less than 4.

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…

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