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Matrices & Determinants
Matrices and Determinants
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Question

For which of the following ordered pairs (μ\mu , δ\delta ), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = μ\mu 4x + 4y + 4z = δ\delta is inconsistent ?

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Solution

This problem asks us to identify for which ordered pair (μ,δ)(\mu, \delta) a given system of linear equations is inconsistent. We will use the conditions for consistency and inconsistency of a system of linear equations, often associated with Cramer's Rule or Gaussian elimination.

Key Concept: Conditions for Inconsistency of a System of Linear Equations

For a system of linear equations Ax=BAx=B, where AA is the coefficient matrix and BB is the constant vector:

  1. Calculate the determinant of the coefficient matrix, Δ=det(A)\Delta = \det(A).
    • If Δ0\Delta \neq 0, the system has a unique solution (consistent).
    • If Δ=0\Delta = 0, the system either has infinitely many solutions (consistent) or no solution

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