Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The value(s) of $\lambda$ for which the system of equations<br/>$(1-\lambda)x + 3y - 4z = 0$<br/>$x - (3+\lambda)y + 5z = 0$<br/>$3x + y - \lambda z = 0$<br/>possesses non-trivial solutions:</p>
<p>(a) $-1$</p>
<p>(b) $0$</p>
<p>(c) $1$</p>
<p>(d) $2$</p>
Step-by-Step Solution
Key Concept: A homogeneous system has non-trivial solutions if and only if the coefficient determinant equals zero.
<p>For non-trivial solutions to exist, the coefficient determinant must be zero:<br/>$\begin{vmatrix} 1-\lambda & 3 & -4 \\ 1 & -(3+\lambda) & 5 \\ 3 & 1 & -\lambda \end{vmatrix} = 0$<br/>Expand and solve for $\lambda$.</p>
Correct Answer: a, b, c