Let A be a `3xx3` matrix given by ` A = [a_(ij)].` If for every
column vector `X, X^(T)AX=O and a_(23)=-1008, ` the sum
of the digits of `a_(32)` is

Answer

9

Explanation

Let `X= [[x_(1)],[x_(2)],[x_(3)]]` and given `X^(T) AX = O`
`rArr [x_(1) x_(2) x_(3)] [[a_(11), a_(12), a_(13)],[a_(21),a_(22),a_(23)],[a_(31), a_(32), a_(33) ]][[x_(1)],[x_(2)],[x_(3)]]= O`
`rArr [x_(1) x_(2) x_(3)] [[a_(11)x_(1)+ a_(12)x_(2)+ a_(13)x_(3)],[a_(21)x_(1)+a_(22)x_(2)+a_(23)x_(3)],[a_(31)x_(1)+ a_(32)x_(2)+ a_(33)x_(3) ]]= O`
` a_(11)x_(1)^(2)+ a_(12)x_(1)x_(2)+ a_(13)x_(1)x_(3)+a_(21)x_(1)x_(2)+a_(22)x_(2)^(2)+a_(23)x_(2)x_(3)`
`+a_(31)x_(1) x_(3) + a_(32)x_(2)x_(3)+a_(33) x_(3)^(2)= 0`
`rArr a_(11) x_(1)^(2) + a_(22) x_(2) ^(2) + a_(33) x_(3) ^(2) + (a_(12)+ a_(21))x_(1) x_(2) + (a_(23) + a_(32)) x_(2) x_(3) + (a_(31)+a_(13)) x_(3) x_(1) = 0`
it is true for every `x_(1), x_(2), x_(3)` then
`a_(11) = a_(22) =a_(33) = 0 and a_(12) = -a_(21), a_(23)=-a_(32),a_13=-a_(31)`
Now , as `a_(23) = - 1008 rArr a_(32) = 1008` `therefore ` Sum of digits `= 1+ 0 + 0 + 8 =9`