Matrices & Determinants
Systems of Linear Equations
Grade 12
Question:
<p>The system of equations <i>x</i> + <i>y</i> + <i>z</i> = 5; <i>x</i> + 2<i>y</i> + 3<i>z</i> = 9; <i>x</i> + 3<i>y</i> + <i>D</i><i>z</i> = <i>I</i> is called smart, if it has a solution. The condition for this is</p>
<p>(a) <i>D</i> ≠ 5 or <i>D</i> = 5 and <i>I</i> = 13</p>
<p>(b) <i>D</i> ≠ 5 and <i>I</i> = 13</p>
<p>(c) <i>D</i> ≠ 5 and <i>I</i> ≠ 13</p>
<p>(d) <i>D</i> ≠ 5 or <i>D</i> = 5 and <i>I</i> ≠ 13</p>
Step-by-Step Solution
Key Concept: Apply consistency conditions for systems of linear equations: unique solution when determinant ≠ 0, or check rank when determinant = 0.
<p><strong>Solution:</strong> For the system to have a solution (smart), the consistency condition must be satisfied. If <i>D</i> ≠ 5, the system always has a unique solution. If <i>D</i> = 5, the system has a solution only when <i>I</i> = 13 (infinitely many solutions). Thus the condition is <i>D</i> ≠ 5 or (<i>D</i> = 5 and <i>I</i> = 13).</p>
Correct Answer: a