The least value of the product xyz for
which the determinant

is

non-negative, is :

  • Option 1)

    -2\sqrt{2}

  • Option 2)

    -16\sqrt{2}

  • Option 3)

    -8

  • Option 4)

    -1

 

Answers (1)

As we learnt in 

Value of determinants of order 3 -

-

 

 \begin{vmatrix} x & 1 & 1\\ 1 & y & 1\\ 1 & 1 & z \end{vmatrix}\geqslant 0

\because x\left ( yz-1 \right ) - 1\left ( z-1 \right )+ 1 \left ( 1-y \right )\geqslant 0

\therefore xyz - x-y-z+2\geqslant 0

\because xyz+z\geqslant x+y+z

So that minimum value of x, y, z are: x = -2, y = -2 and z = -2 which satisfy this inequality.

\therefore xyz= -2\times -2\times -2=-8


Option 1)

-2\sqrt{2}

This option is incorrect.

Option 2)

-16\sqrt{2}

This option is incorrect.

Option 3)

-8

This option is correct.

Option 4)

-1

This option is incorrect.

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