Matrices & Determinants
System of linear equations
Grade 12
Question:
<p><strong>For Problems 22–24</strong><br>Consider the system of equations<br>\(x + y + z = 6\)<br>\(x + 2y + 3z = 10\)<br>\(x + 2y + \lambda z = \mu\)<br>The system has unique solution if</p>
<p>\(\lambda \neq 3\)</p>
<p>\(\lambda = 3,\ \mu = 10\)</p>
<p>\(\lambda = 3,\ \mu \neq 10\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: A system of linear equations has a unique solution if and only if the coefficient matrix is non-singular (determinant ≠ 0). You must find the condition on λ that makes det(A) ≠ 0.
<p><strong>Step 1:</strong> Write the coefficient matrix $A$:</p>
<p>$$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \lambda \end{bmatrix}$$</p>
<p><strong>Step 2:</strong> Calculate $\det(A)$ by expanding along Row 1:</p>
<p>$$\det(A) = 1(2\lambda - 6) - 1(\lambda - 3) + 1(2 - 2)$$</p>
<p>$$\det(A) = 2\lambda - 6 - \lambda + 3 + 0$$</p>
<p>$$\det(A) = \lambda - 3$$</p>
<p><strong>Step 3:</strong> For a unique solution, we need $\det(A) \neq 0$:</p>
<p>$$\lambda - 3 \neq 0$$</p>
<p>$$\lambda \neq 3$$</p>
<p><strong>Step 4:</strong> Note that $\mu$ can be any real number when $\lambda \neq 3$. The system has a unique solution for all values of $\mu$ if $\lambda \neq 3$.</p>
<p><strong>Answer:</strong> The system has a unique solution if $\lambda \neq 3$ (and $\mu \in \mathbb{R}$)</p>
Correct Answer: A