Let `A` be a `3xx3` matrix given by `A=(a_(ij))_(3xx3)`. If for every column vector `X` satisfies `X'AX=0` and `a_(12)=2008`, `a_(13)=2010` and `a_(23)=-2012`. Then the value of `a_(21)+a_(31)+a_(32)=`

Options

A. `-6`
B. `2006`
C. `-2006`
D. `0`

Answer

C

Explanation

`(c )` Let `X=[{:(x_(1)),(x_(2)),(x_(3)):}]`
`({:(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)):})=0`
`a_(11)x_(1)^(2)+a_(22)X_(2)^(2)+a_(33)x_(3)^(2)+(a_(12)+a_(21))x_(1)x_(2)+(a_(13)+a_(31))x_(1)x_(3)+(a_(23)+a_(32))x_(2)x_(3)=0`
It is true for every `x_(1)`, `x_(2)`, `x_(3)`
then `a_(11)=a_(22)=a_(33)=0`,`a_(12)+a_(21)=0`, `a_(13)+a_(31)=0` ,`a_(23)+a_(32)=0`
`:.A` is a skew symmetric matrix
`a_(21)=-2008`
`a_(31)=-2010`
`a_(32)=2012`