Matrices & Determinants
System of Linear Equations and Consistency
Grade 12
Question:
<p>If the system of equations<br/>$ax + y + 2z = 0$<br/>$x + 2y + z = b$<br/>$2x + y + az = 0$<br/>has no solution, then $(a+b)$ can be equal to:</p>
<p>(a) $-1$</p>
<p>(b) $2$</p>
<p>(c) $3$</p>
<p>(d) $4$</p>
Step-by-Step Solution
Key Concept: A system has no solution when the coefficient determinant is zero but the augmented matrix has higher rank.
<p>For the system to have no solution, the coefficient determinant must be zero (inconsistent system):<br/>$\begin{vmatrix} a & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & a \end{vmatrix} = 0$<br/>and the system must be inconsistent. Expand to find conditions on $a$, then use consistency conditions to find $b$.</p>
Correct Answer: c