Question

Let for $i=1,2,3,p_i\left(x\right)$ be a polynomial of degree 2 in x, $p_i^{\prime }\left(x\right)$ and $p_i^{\prime \prime }\left(x\right)$ be the first and second order derivatives of $p_i\left(x\right)$ respectively. Let,
$A\left(x\right)=\left[\begin{array}{ccc}{p}_1\left(x\right)& p_1^{\prime }\left(x\right)& p_1^{\prime \prime }\left(x\right)\\ p_2\left(x\right)& p_2^{\prime }\left(x\right)& p_2^{\prime \prime }\left(x\right)\\ p_3\left(x\right)& p_3^{\prime }\left(x\right)& p_3^{\prime \prime }\left(x\right)\end{array}\right]$
and $B\left(x\right)={\left[A\left(x\right)\right]}^{T}A\left(x\right).$ Then determinant of $B\left(x\right)$ :

• A. is a polynomial of degree $6$ in $x$.
• B. is a polynomial of degree $3$ in $x$.
• C. is a polynomial of degree $2$ in $x$.
• D. does not depend on $x$.

Answer 1

Answer: (A)

Let $P_i=a_i{x}^2+b_{i}X+C_i{a}_i\ne 0$
$b_i,c_i\in R$
$A\left(X\right)=\left[\begin{array}{ccc}{a}_1{x}^2+b_{1}x+c_1& 2a_{1}x+b_1& 2a_1\\ a_2{x}^2+b_{2}x+c_2& 2a_{2}x+b_2& 2a_2\\ a_3{x}^2+b_{3}x+c_3& 2a_{3}x+b_3& 2a_3\end{array}\right]$
use $\left(i\right)C_2\to C_2-x{C}_3$
then use $\left(ii\right)C_1\to C_1-x{C}_2-\frac{x^2}{2}{C}_3$
$A\left(X\right)=\left[\begin{array}{ccc}{c}_1& b_1& 2a_1\\ c_2& b_2& 2a_2\\ c_3& b_3& 2a_3\end{array}\right]\Rightarrow \left|A\right|=$ constant
So $\left|B\right|=\left|A^T\right|\left|A\right|={\left|A\right|}^2=$ constant independent from n