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 good, if it has infinitely many solutions. 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, <i>I</i> is any real number</p>
Step-by-Step Solution
Key Concept: Infinitely many solutions occur when the system is dependent and consistent, requiring both the coefficient determinant to be 0 and the consistency condition to hold.
<p><strong>Solution:</strong> For infinitely many solutions, the coefficient matrix and augmented matrix must have the same rank but less than 3. This occurs when <i>D</i> = 5 (making the third row dependent on the first two) and <i>I</i> = 13 (maintaining consistency).</p>
Correct Answer: b