Matrices & Determinants
Determinant Identities
Grade 12
Question:
<p>If \(f(x), g(x)\) and \(h(x)\) are polynomials of degree 4 and \(\begin{vmatrix} f(x) & g(x) & h(x) \\ a & b & c \\ p & q & r \end{vmatrix} = mx^4 + nx^3 + rx^2 + sx + t\) be an identity in \(x\), then \(\frac{f'''(0) - f''(0)}{a} + \frac{g'''(0) - g''(0)}{b} + \frac{h'''(0) - h''(0)}{c}\) is equal to</p>
<p>(a) \(2^3(n - r)\)</p>
<p>(b) \(3^2(n + r)\)</p>
<p>(c) \(32(n - r)\)</p>
<p>(d) \(23(n + r)\)</p>
Step-by-Step Solution
Key Concept: Apply the derivative rule to determinants and evaluate at specific points to extract coefficient relationships.
<p>Differentiate the determinant identity three times and evaluate at $x = 0$. The polynomial $mx^4 + nx^3 + rx^2 + st + t$ gives derivatives involving coefficients. Extracting the coefficient relationship through the Leibniz rule for derivatives of determinants yields $32(n - r)$.</p>
Correct Answer: C