Mathematics - Matrices Question with Solution | TestHub

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. $$\begin{array}{c}\therefore \mathrm{A}(x)=\left[\begin{array}{ccc}{a}_1{x}^2+b_{1}x+c_1& 2a_{1}x+b_1& 2a_1\\ a_2{x}^2+b_{2}x+c_2& 2a_{2}x+b_2& 2a_2\\ a_3{x}^2+b_{3}x+c_3& 2a_{3}x+b_3& 2a_3\end{array}\right]\\ =\left[\begin{array}{lll}{a}_1{x}^2+b_{1}x+c_1& a_2{x}^2+b_{2}x+c_2& a_3{x}^2+b_{3}x+c_3\\ 2a_{1}x+b_1& 2a_{2}x+b_2& 2a_{3}x+b_2\\ 2a_1& 2a_2& 2a_3\end{array}\right]\\ \times \left[\begin{array}{ccc}{a}_1{x}^2+b_{1}x+c_1& 2a_{1}x+b_1& 2a_1\\ a_2{x}^2+b_{2}x+c_2& 2a_{2}x+b_2& 2a_2\\ a_3{x}^2+b_{3}x+c_3& 2a_{3}x+b_3& 2a_3\end{array}\right]\end{array}$$ It is clear from the above multiplication, the degree of determinant of $\mathrm{B}(x)$ can not be less than 4 .