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 lazy, if it has no 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: An inconsistent system occurs when the coefficient determinant is 0 but the consistency condition is violated.
<p><strong>Solution:</strong> For no solution (inconsistent system), the coefficient matrix rank must be less than the augmented matrix rank. This occurs when <i>D</i> = 5 (third row becomes dependent on first two) but <i>I</i> ≠ 13 (inconsistency in the augmented column).</p>
Correct Answer: c