Let $A=\begin{bmatrix} x & y & z \ y & z & x \ z & x & y \end{bmatrix}$, 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 $A=\begin{bmatrix} x & y & z \ y & z & x \ z & x & y \end{bmatrix}$, 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

Aicharya · Accepted Answer

A 2 = I &#8658; A A ' = I (as A ' = A ) &#8658; A is orthogonal So, x 2 + y 2 + z 2 = 1 and x y + y z + z x = 0 &#8658; x + y + z 2 = 1 + 2 &#215; 0 &#8658; x + y + z = 1 Thus, x 3 + y 3 + z 3 = 3 &#215; 2 + 1 &#215; 1 - 0 = &#160; 7

Let $A=\begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$, where $x$, $y$ and $z$ are real…

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