Matrices & Determinants
Systems of Linear Equations
Grade 12
Question:
<p>Number of values of <i>a</i> for which the system of equations <i>ax</i> + (2 − <i>a</i>)<i>y</i> = 4 + <i>a</i> and <i>ax</i> + (2<i>a</i> − 1)<i>y</i> = <i>a</i>² − 2 possess no solution, is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) infinite</p>
Step-by-Step Solution
Key Concept: A system of linear equations has no solution when the coefficient determinant equals zero but the system is inconsistent (rank of coefficient matrix ≠ rank of augmented matrix).
<p><strong>Solution:</strong> For the system to have no solution, the coefficient determinant must be zero while the augmented determinant is non-zero.</p><p>The coefficient determinant is:</p><p>$$\Delta = \begin{vmatrix} a & 2-a \\ a & 2a-1 \end{vmatrix} = a(2a-1) - a(2-a) = 2a^2 - a - 2a + a^2 = 3a^2 - 3a = 3a(a-1)$$</p><p>For no solution: $\Delta = 0$ when $a = 0$ or $a = 1$</p><p>We need to verify these values make the system inconsistent by checking the augmented determinants are non-zero.</p><p>∴ Answer is (c) 2.</p>
Correct Answer: c