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Question

If $A equals left square bracket x y z right square bracket comma B equals open square brackets table attributes columnspacing 1em end attributes row cell a straight h invisible function application g end cell row cell h b f end cell row cell g f c end cell end table close square brackets$ and $C equals open square brackets table attributes columnalign left columnspacing 1em end attributes row x row y row z end table close square brackets$ Then, $B to the power of straight prime left parenthesis A B right parenthesis equals 0$ if

$[a x^{2} + b y^{2} + c z^{2} + 2 g x y + 2 f y z + 2 c z x] = 0$
$[a x^{2} + c y^{2} + b z^{2} + x y + y z + z x] = 0$
$[a x^{2} + b y^{2} + c z^{2} + 2 h x y + 2 b y + 2 c z] = 0$
$[a x^{2} + b y^{2} + c z^{2} + 2 g z x + 2 h x y + 2 f y z] = 0$

The correct answer is: $open square brackets a x squared plus b y squared plus c z squared plus 2 g z x plus 2 h x y plus 2 f y z close square brackets equals 0$

$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell A B equals left square bracket x y z right square bracket open square brackets table attributes columnalign center columnspacing 1em end attributes row cell a h g end cell row cell h b f end cell row cell g f c end cell end table close square brackets end cell row cell equals left square bracket x a plus y h plus z g x h plus y b plus z f x g plus y f plus z c right square bracket end cell end table$
Now, ABC = 0
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow left square bracket x a plus y h plus z g x h plus y b plus z f x g plus y f plus z c right square bracket open square brackets table attributes columnspacing 1em end attributes row x row y row z end table close square brackets equals 0 end cell row cell not stretchy rightwards double arrow open square brackets a x squared plus h x y plus g x z plus h x y plus y squared b plus f z y plus g x z plus y f z plus z squared c close square brackets end cell row cell equals O end cell row cell not stretchy rightwards double arrow open square brackets a x squared plus b y squared plus c z squared plus 2 g z x plus 2 h x y plus 2 f y z close square brackets equals O end cell end table$

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