Is Cramer's Rule applicable for solving the linear system below? I

Question

Is Cramer's Rule applicable for solving the linear system below? If yes, apply it. Otherwise, alter the last equation in the system so that the solution can be obtained by applying the Rule.

$x+y+z=\pi$

$-\pi x+\pi y+\sqrt{2}z=0$

$\pi^2 x+\pi^2 y+2z=0.$

26 May 2025

Answer :

Word Count : 742

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 ____ _____ _________ __________ ________ _______ _______ ___ ______.
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