"Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real numbers such that x + y + z 0 and xyz = 2. If ${A^2} = {I_3}$, then the value of ${x^3} + {y^3} + {z^3}$ is ____________."

Question

"Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If ${A^2} = {I_3}$, then the value of ${x^3} + {y^3} + {z^3}$ is ____________."

Accepted Answer

"$$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$

$\therefore$ $|A| = \left( {{x^3} + {y^3} + {z^3} - 3xyz} \right)$

Given ${A^2} = {I_3}$

$|{A^2}| = 1$

$\therefore$ ${({x^3} + {y^3} + {z^3} - 3xyz)^2} = 1$

$\Rightarrow {x^3} + {y^3} + {z^3} - 3xyz = 1$ only as $(x + y + z > 0)$

$\Rightarrow {x^3} + {y^3} + {z^3} = 6 + 1 = 7$"

Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If ${A^2} = {I_3}$, then the value of ${x^3} + {y^3} + {z^3}$ is ____________.

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Let $$A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]$$, where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If ${A^2} = {I_3}$, then the value of ${x^3} + {y^3} + {z^3}$ is ____________.

Solution

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