1. Key Concepts and Formulas

This problem primarily utilizes the following key concepts and formulas:

• Determinant of a Matrix: Specifically, the method to calculate the determinant of a $3 \times 3$ matrix. For a matrix $M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$, its determinant is $|M| = a(ei - fh) - b(di - fg) + c(dh - eg)$.
• Properties of Determinants:
  • The determinant of a product of matrices is the product of their determinants: $|AB| = |A||B|$.
  • The determinant of an identity matrix $I_n$ is $1$: $|I_n| = 1$.
• Algebraic Identity: The factorization of the sum of cubes minus three times their product: $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$
• Completing the Square for Quadratic Forms: The term $x^2+y^2+z^2-xy-yz-zx$ can be rewritten as: $x^2+y^2+z^2-xy-yz-zx = \frac{1}{2}[(x-y)^2+(y-z)^2+(z-x)^2]$ This form clearly shows that for real numbers $x, y, z$, this expression is always non-negative, i.e., $x^2+y^2+z^2-xy-yz-zx \ge 0$. It is zero only if $x=y=z$.

---

2. Step-by-Step Solution

We are given the matrix A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right] and the conditions $x, y, z \in \mathbb{R}$, $x+y+z > 0$, $xyz = 2$, and $A^2 = I_3$. We need to find the value of $x^3+y^3+z^3$.

Step 1: Calculate the Determinant of Matrix A

First, we calculate the determinant of the given matrix $A$. This is a crucial step because the condition $A^2 = I_3$ involves the determinant. $|A| = x \begin{vmatrix} z & x \\ x & y \end{vmatrix} - y \begin{vmatrix} y & x \\ z & y \end{vmatrix} + z \begin{vmatrix} y & z \\ z & x \end{vmatrix}$ Expanding the $2 \times 2$ determinants: $|A| = x(z \cdot y - x \cdot x) - y(y \cdot y - x \cdot z) + z(y \cdot x - z \cdot z)$ $|A| = x(yz - x^2) - y(y^2 - xz) + z(xy - z^2)$ Distributing the terms: $|A| = xyz - x^3 - y^3 + xyz + xyz - z^3$ Combining like terms, we get: $|A| = 3xyz - (x^3 + y^3 + z^3)$ Explanation: We use the standard cofactor expansion method for a $3 \times 3$ matrix to express its determinant in terms of $x, y, z$. This specific form of matrix (a type of circulant matrix) often leads to this determinant expression.

Step 2: Utilize the Condition $A^2 = I_3$

We are given that $A^2 = I_3$, where $I_3$ is the $3 \times 3$ identity matrix. Taking the determinant of both sides of the equation: $|A^2| = |I_3|$ Using the property that $|A^2| = |A \cdot A| = |A||A| = (|A|)^2$: $(|A|)^2 = 1$ This implies that $|A|$ can be either $1$ or $-1$: $|A| = \pm 1$ Explanation: The property $|A^2| = (|A|)^2$ is fundamental for relating matrix equations to scalar equations involving determinants. Since $I_3$ is the identity matrix, its determinant is always 1. This step allows us to establish a numerical value for $|A|$.

Step 3: Relate Determinant to the Expression and Use Given Conditions

From Step 1, we have $|A| = 3xyz - (x^3 + y^3 + z^3)$. From Step 2, we have $|A| = \pm 1$. Therefore, we can write: $3xyz - (x^3 + y^3 + z^3) = \pm 1$