Question

Let $A$ be a $3\times 3$ matrix given by $A=\left[a_{ij}\right]$.
If for every column vector $X$, $X^{T}AX=0$ and $a_{23}=-1008$,then the sum of the digits of $a_{32}$ is

Answer 1

$X^{T}AX=0$(given)

Let $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

$X^{T}AX={\left(X^{T}AX\right)}^T=X^T{A}^{T}X=-X^{T}AX$

$\Rightarrow X^{T}AX=0$

$\therefore X^T{A}^{T}X=-X^{T}AX$

$\therefore A^T=-A$

$\therefore a_{23}=-a_{32}$ or $a_{32}=-a_{23}$

$\Rightarrow a_{32}=-(-1008)=1008$

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