Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The system of equations <span class="math">x + y + z = 2</span>, <span class="math">2x + y - z = 3</span>, <span class="math">3x + 2y + \lambda z = 4</span> has unique solution if</p>
<p>(a) <span class="math">\lambda \neq 0</span></p>
<p>(b) <span class="math">-1 < \lambda < 1</span></p>
<p>(c) <span class="math">\lambda = 0</span></p>
<p>(d) <span class="math">-2 < \lambda < 2</span></p>
Step-by-Step Solution
Key Concept: A system of linear equations has a unique solution if and only if the determinant of the coefficient matrix is non-zero.
<p>For a system to have a unique solution, the coefficient matrix determinant must be non-zero.</p><p><span class="math">\begin{vmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & \lambda \end{vmatrix} \neq 0</span></p><p>Computing the determinant and setting it equal to zero, we find that the determinant is zero when <span class="math">\lambda = 0</span>. Therefore, for unique solution, <span class="math">\lambda \neq 0</span>.</p>
Correct Answer: a