Discussion
If the following system of linear equations
\(x - 2y + z = 5\)
\(2x - y + 2z = 7\)
\(x + 2y + \lambda z = 5\) has a unique solution, then \(\lambda \neq\)
Answer: A
A unique solution needs \(\lambda \neq 1\).
A linear system has a unique solution exactly when the determinant of its coefficient matrix is non-zero:
\[ \begin{aligned} \begin{vmatrix} 1 & -2 & 1 \\ 2 & -1 & 2 \\ 1 & 2 & \lambda \end{vmatrix} &= 1(-\lambda - 4) + 2(2\lambda - 2) + 1(5) \\ &= 3\lambda - 3 \end{aligned} \]
This is zero only for \(\lambda = 1\), so for a unique solution \(\lambda\) must not be 1. Any of the other listed values gives a non-zero determinant, so they are all allowed.
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