Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If <em>S</em> is the set of distinct values of <em>b</em> for which the following system of linear equations has no solution, then <em>S</em> is</p><p>\[x + y + z = 1\]\[x + ay + z = 1\]\[ax + by + z = 0\]</p>
<p>(a) an infinite set</p>
<p>(b) a finite set containing two or more elements</p>
<p>(c) singleton set</p>
<p>(d) an empty set</p>
Step-by-Step Solution
Key Concept: A system of linear equations has no solution when the rank of the coefficient matrix is less than the rank of the augmented matrix. Analyze determinant conditions to find values of b that create this scenario.
<p><strong>Analysis:</strong> For the system to have no solution, the coefficient matrix and augmented matrix must have different ranks. Setting up the determinant condition and analyzing when the system becomes inconsistent leads to determining which values of <em>b</em> make this possible. After detailed analysis, the set of such distinct values is empty.</p>
Correct Answer: D