Matrices & Determinants
System of linear equations - unique solution
Grade 12
Question:
<p>Let <em>S</em> be the set of all real values of <em>k</em> for which the system of linear equations<br>\(x + y + z = 2\)<br>\(2x + y - z = 3\)<br>\(3x + 2y + kz = 4\)<br>has a unique solution. Then <em>S</em> is</p>
<p>an empty set</p>
<p>equal to \(\{0\}\)</p>
<p>equal to \(R\)</p>
<p>equal to \(R - \{0\}\)</p>
Step-by-Step Solution
Key Concept: A system of 3 linear equations has a unique solution if and only if the coefficient matrix is non-singular (determinant ≠ 0). Find the determinant and set it ≠ 0 to find all valid values of k.
<p><strong>Step 1:</strong> Write the coefficient matrix $A$:</p>
<p>$$A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & k \end{bmatrix}$$</p>
<p><strong>Step 2:</strong> Calculate $\det(A)$ using first row expansion:</p>
<p>$$\det(A) = 1(k + 2) - 1(2k + 3) + 1(4 - 3)$$
$$= k + 2 - 2k - 3 + 1$$
$$= -k$$</p>
<p><strong>Step 3:</strong> For a unique solution, we need $\det(A) \neq 0$:</p>
<p>$$-k \neq 0$$
$$k \neq 0$$</p>
<p><strong>Step 4:</strong> Therefore, $S = \mathbb{R} \setminus \{0\}$ or equivalently $S = (-\infty, 0) \cup (0, \infty)$</p>
<p>$\therefore$ Answer: $S$ is all real numbers except $k = 0$ (Option D)</p>
Correct Answer: D