Question

If $A=\left[xyz\right],B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ and $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$
Then, $ABC=0$, if

• A. $[a{x}^2+b{y}^2+c{z}^2+2gxy+2fyz+2czx]=0$
• B. $[a{x}^2+c{y}^2+b{z}^2+xy+yz+zx]=0$
• C. $[a{x}^2+b{y}^2+c{z}^2+2hxy+2by+2cz]=0$
• D. $[a{x}^2+b{y}^2+c{z}^2+2gzx+2fyz+2hxy]=0$

Answer 1

Answer: (D)

$[a{x}^2+b{y}^2+c{z}^2+2gzx+2fyz+2hxy]=0$

$A=\left[xyz\right]B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]c=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$A.B=\left[xyz\right]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$

$=[ax+hy+gzhx+by+fzgx+fy+cz]$

$A.B.C=\left(AB\right)\times \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=[a{x}^2+hyx+gyx+hxy+b{y}^2+fzy+gxz+fyz+c{z}^2]$

$ABC=[a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fzy]$

$ABC=0$

when,

$a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fzy=0$