A system of equations $\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & -4 \\ 1 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix}$ is
Options:
(A) consistent and has unique solution
(B) inconsistent and has no solution
(C) consistent and has infinite solutions
(D) inconsistent and has unique solution
Official Answer Key:
A
Step-by-Step Solution & Explanation
det(A) = 1(1×1−(−4)×3) − (−1)(2×1−(−4)×1) + 2(2×3−1×1) = 1(13) + 1(6) + 2(5) = 29 ≠ 0. A non-zero determinant guarantees the coefficient matrix is invertible, so the system has exactly one solution (x ≈ 2.241, y ≈ −0.034, z ≈ 0.862 by Cramer's rule/Gaussian elimination). 'Inconsistent' options are ruled out immediately once det ≠ 0.
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