Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The system of linear equations<br/>\(x + λy - z = 0\)<br/>\(λx + y - z = 0\)<br/>\(x + y - λz = 0\)<br/>has a non-trivial solution for</p>
<p>(a) exactly one value of \(λ\)</p>
<p>(b) exactly two values of \(λ\)</p>
<p>(c) exactly three values of \(λ\)</p>
<p>(d) infinitely many values of \(λ\)</p>
Step-by-Step Solution
Key Concept: Set the coefficient determinant to zero and solve for the parameter.
<p><strong>Solution:</strong> For non-trivial solutions, the determinant of the coefficient matrix must be zero. Calculate:<br/>$\begin{vmatrix} 1 & λ & -1 \\ λ & 1 & -1 \\ 1 & 1 & -λ \end{vmatrix} = 0$<br/>This gives a polynomial equation in $λ$ with two solutions.</p>
Correct Answer: B