Calculate and factor the determinant.

01 / Standard#Your turn

C=\begin{pmatrix}a+b&b+c&c+a\\a-b&b-c&c-a\\c&a&b\end{pmatrix}

Official paper · jm02-2025 · 5(a)(i) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand along the first row, or combine rows first.

Hint 2

Use the factorisation of a³+b³+c³−3abc.

Worked solution

Expansion and collection leave a symmetric cubic.
$|C|=2(a^3+b^3+c^3-3abc)$
Factor and rewrite the quadratic factor as a sum of squares.
$|C|=(a+b+c)\big[(a-b)^2+(b-c)^2+(c-a)^2\big]$

|C|=(a+b+c)[(a−b)²+(b−c)²+(c−a)²].

Checks and common pitfalls: A row interchange would reverse the determinant sign.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expansion and collection leave a symmetric cubic.
$|C|=2(a^3+b^3+c^3-3abc)$
Factor and rewrite the quadratic factor as a sum of squares.
$|C|=(a+b+c)\big[(a-b)^2+(b-c)^2+(c-a)^2\big]$

Think first. Reveal a hint when the class is ready.

2025 JM02