Matrices & Determinants
Determinant Expansion and Polynomial Coefficients
Grade 12
Question:
<p>Let $D(x) = \begin{vmatrix} x^2+4x-3 & 2x+4 & 13 \\ 2x^2+5x-9 & 4x+5 & 26 \\ 8x^2-6x+1 & 16x-6 & 104 \end{vmatrix} = ax^3 + bx^2 + \gamma x + d$, then:</p>
<p>(a) $a + b = 0$</p>
<p>(b) $b + \gamma = 0$</p>
<p>(c) $a + b + \gamma + d = 0$</p>
<p>(d) $a + b + \gamma = 0$</p>
Step-by-Step Solution
Key Concept: Expand the determinant to obtain a polynomial and verify coefficient relationships; check special values.
<p>Expand the determinant as a polynomial in $x$. Find coefficients $a, b, \gamma, d$ and verify the relations among them. Note that $D(1) = 0$ gives $a+b+\gamma+d=0$, and other relations can be verified by expansion.</p>
Correct Answer: a, c, d