Let A be a 3 x 3  matrix given by A =[a_ij] . if for every column vector  X, .X^T AX =O and a₂₃ =-1008 , then sum of the digits of a₃₂ is

Let $\mathrm{X}=\left[\begin{array}{l}{\mathrm{x}}_1\\ {\mathrm{x}}_2\\ {\mathrm{x}}_3\end{array}\right]$ and given ${\mathrm{X}}^{\mathrm{T}}\mathrm{AX}=\mathrm{O}$

$\begin{array}{l}\Rightarrow \left[\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{x}}_2& {\mathrm{x}}_3\end{array}\right]\left[\begin{array}{lll}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right]\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\\ {\mathrm{x}}_3\end{array}\right]=\mathrm{O}\\ \Rightarrow \left[\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{x}}_2& {\mathrm{x}}_3\end{array}\right]\left[\begin{array}{l}{\mathrm{a}}_{11}{\mathrm{x}}_1+{\mathrm{a}}_{12}{\mathrm{x}}_2+{\mathrm{a}}_{13}{\mathrm{x}}_3\\ {\mathrm{a}}_{21}{\mathrm{x}}_1+{\mathrm{a}}_{22}{\mathrm{x}}_2+{\mathrm{a}}_{23}{\mathrm{x}}_3\\ {\mathrm{a}}_{31}{\mathrm{x}}_1+{\mathrm{a}}_{32}{\mathrm{x}}_2+{\mathrm{a}}_{33}{\mathrm{x}}_3\end{array}\right]=\mathrm{O}\\ \Rightarrow {\mathrm{a}}_{11}{\mathrm{x}}_1^2+{\mathrm{a}}_{12}{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{a}}_{13}{\mathrm{x}}_1{\mathrm{x}}_3+{\mathrm{a}}_{21}{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{a}}_{22}{\mathrm{x}}_2^2+{\mathrm{a}}_{23}{\mathrm{x}}_2{\mathrm{x}}_3\Rightarrow {\mathrm{a}}_{31}{\mathrm{x}}_1{\mathrm{x}}_3+{\mathrm{a}}_{32}{\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{a}}_{33}{\mathrm{x}}_3^2=0\\ \\ \Rightarrow {\mathrm{a}}_{22}{\mathrm{x}}_2^2+{\mathrm{a}}_{33}{\mathrm{x}}_3^2+\left({\mathrm{a}}_{12}+{\mathrm{a}}_{21}\right){\mathrm{x}}_1{\mathrm{x}}_2+\left({\mathrm{a}}_{23}+{\mathrm{a}}_{32}\right){\mathrm{x}}_2{\mathrm{x}}_3+\left({\mathrm{a}}_{31}+{\mathrm{a}}_{13}\right){\mathrm{x}}_3{\mathrm{x}}_1=0\\ \end{array}$

it is true for every x₁, x₂ x₃, then

 ${\mathrm{a}}_{11}={\mathrm{a}}_{22}={\mathrm{a}}_{33}=0$ and  ${\mathrm{a}}_{12}=-{\mathrm{a}}_{21},{\mathrm{a}}_{23}=-{\mathrm{a}}_{32},{\mathrm{a}}_{13}=-{\mathrm{a}}_{31}$

now as ${\mathrm{a}}_{23}=-1008\Rightarrow {\mathrm{a}}_{32}=1008$

sum of digits = 1 +0+0+8 =9