Word Count : 742

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We are given the system of equations: 1. $x + y + z = \pi$ 2. $-\pi x + \pi y + \sqrt{2}z = 0$ 3. $\pi^2 x + \pi^2 y + 2z = 0$ --- ### Step 1: Write in matrix form Let the system be $AX = B$, where: $$ A = \begin{bmatrix} 1 & 1 & 1 \\ -\pi & \pi & \sqrt{2} \\ \pi^2 & \pi^2 & 2 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} \pi \\ 0 \\ 0 \end{bmatrix} $$ --- ### Step 2: Check applicability of Cramer’s Rule Cramer’s Rule can be applied only if the coefficient matrix $A$ is invertible, i.e., $\det(A) \neq 0$. We compute the determinant of $A$: $$ \det(A) = \begin{vmatrix} 1 & 1 & 1 \\ -\pi & \pi & \sqrt{2} \\ \pi^2 & \pi^2 & 2 \end{vmatrix} _____ _____ ________ _______ ________ __________.
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