Matrices & Determinants
Determinant factorization
Grade 12
Question:
<p>If \(2s = a + b + c\) and the determinant \(\begin{vmatrix} a^2 & (s-a)^2 & (s-a)^2 \\ (s-b)^2 & b^2 & (s-b)^2 \\ (s-c)^2 & (s-c)^2 & c^2 \end{vmatrix} = ks^3(s-a)(s-b)(s-c)\), then the numerical quantity \(k\) should be</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Use row and column operations to simplify the determinant and match it with the given form to find k.
<p>Expand the given determinant using standard methods. Apply row and column operations to simplify. The result equals \(4s^3(s-a)(s-b)(s-c)\), so \(k = 4\).</p>
Correct Answer: D