A homogeneous polynomial of the second d...

A homogeneous polynomial of the second degree in `n` variables i.e., the expression
`phi=sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j)` where `a_(ij)=a_(ji)` is called a quadratic form in `n` variables `x_(1),x_(2)`….`x_(n)` if `A=[a_(ij)]_(nxn)` is
a symmetric matrix and `x=[{:(x_(1)),(x_(2)),(x_(n)):}]` then
`X^(T)AX=[X_(1)X_(2)X_(3) . . . . .X_(n)][{:(a_(11),a_(12) ....a_(1n)),(a_(21),a_(22)....a_(2n)),(a_(n1),a_(n2)....a_(n n)):}][{:(x_(1)),(x_(2)),(x_(n)):}]`
`=sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j)=phi`
Matrix A is called matrix of quadratic form `phi`.
Q. The quadratic form of matrix `A[{:(0,2,1),(2,3,-5),(1,-5,8):}]` is

`3x_(2)^(2)+8x_(3)^(2)+2x_(1)x_(2)+x_(1)x_(3)-5x_(2)x_(3)`

`3x_(2)^(2)+8x_(3)^(2)+4x_(1)x_(2)+2x_(1)x_(3)-10x_(3)x_(2)`

`x_(1)^(2)+2x_(2)^(2)+x_(3)^(2)+3x_(1)x_(2)-5x_(2)x_(3)+8x_(1)x_(2)`

`3x_(1)^(2)+8x_(2)^(2)+4x_(1)x_(2)+2x_(1)x_(3)+10x_(3)x_(2)`

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The correct Answer is:

`=[x_(1)x_(2)x_(3)][{:(2x_(2)+x_(3)),(2x_(1)+3x_(2)-5x_(3)),(x_(1)-5x_(2)+8x_(3)):}]`
`=2x_(1)x_(2)+x_(1)x_(3)+2x_(1)x_(2)+3x_(2)^(2)-5x_(3)x_(2)+x_(3)x_(1)+5x_(3)x_(2)+8x_(3)^(2)`
`=3x_(2)^(3)+8x_(3)^(2)+4x_(1)x_(2)+2x_(1)x_(3)-10x_(2)x_(3)`

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