Matrices & Determinants
Determinants and Inequalities
Grade 12
Question:
<p>If the value of the determinant <span style='display:inline-block; border: 1px solid black; padding: 5px;'>\[\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}\]</span> is positive, then (for \(a, b, c > 0\))</p>
<p>(a) \(abc > 1\)</p>
<p>(b) \(abc > -8\)</p>
<p>(c) \(abc < -8\)</p>
<p>(d) \(abc > -2\)</p>
Step-by-Step Solution
Key Concept: Expand the determinant and apply inequalities (AM-GM) for positive numbers to establish the constraint on $abc$.
<p>Expand the determinant: $$\Delta = a(bc-1) - 1(c-1) + 1(1-b) = abc - a - c + 1 - b + 1 = abc - (a+b+c) + 2$$ For $\Delta > 0$ and $a, b, c > 0$, by AM-GM inequality, $a + b + c \geq 3\sqrt[3]{abc}$. Setting this condition with $\Delta > 0$ yields $abc > 1$.</p>
Correct Answer: A