Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If the system of linear equations \(x + y + z = 6\), \(x + 2y + 3z = 14\), and \(2x + 5y + \lambda z = \mu\) (\(\lambda, \mu \in \mathbb{R}\)) has a unique solution, then</p>
<p>(1) \(\lambda \neq 8\)</p>
<p>(2) \(\lambda = 8, \mu \neq 36\)</p>
<p>(3) \(\lambda = 8, \mu = 36\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: A system of 3 linear equations in 3 unknowns has a unique solution if and only if the coefficient matrix is non-singular (determinant ≠ 0). Calculate det(A) and find the condition on λ that ensures det(A) ≠ 0.
<p><strong>Step 1:</strong> Write the coefficient matrix $A$:</p><p>$$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 2 & 5 & \lambda \end{bmatrix}$$</p><p><strong>Step 2:</strong> Calculate $\det(A)$ by expanding along the first row:</p><p>$$\det(A) = 1(2\lambda - 15) - 1(\lambda - 6) + 1(5 - 4)$$</p><p>$$\det(A) = 2\lambda - 15 - \lambda + 6 + 1$$</p><p>$$\det(A) = \lambda - 8$$</p><p><strong>Step 3:</strong> For unique solution, $\det(A) \neq 0$:</p><p>$$\lambda - 8 \neq 0$$</p><p>$$\therefore \lambda \neq 8 \text{ (independent of } \mu)$$</p><p>The value of $\mu$ does not affect uniqueness; only $\lambda$ matters.</p>
Correct Answer: A