Factor the determinant.

01 / Standard#Your turn

D=\begin{vmatrix}a&b&c\\b+c&c+a&a+b\\1+b&1+c&1+a\end{vmatrix}

Official paper · jm02-2024 · 5(a) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Add row 1 to row 2.

Hint 2

Factor a+b+c and then subtract column 1 from columns 2 and 3.

Worked solution

The second row becomes constant.
$D=(a+b+c)\begin{vmatrix}a&b&c\\1&1&1\\1+b&1+c&1+a\end{vmatrix}$
Column differences make expansion along row 2 simple.
$D=-(a+b+c)\begin{vmatrix}b-a&c-a\\c-b&a-b\end{vmatrix}$
Expand and collect the quadratic factor.
$D=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

(a+b+c)(a²+b²+c²−ab−bc−ca).

Checks and common pitfalls: The cofactor in position (2,1) carries a minus sign.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The second row becomes constant.
$D=(a+b+c)\begin{vmatrix}a&b&c\\1&1&1\\1+b&1+c&1+a\end{vmatrix}$
Column differences make expansion along row 2 simple.
$D=-(a+b+c)\begin{vmatrix}b-a&c-a\\c-b&a-b\end{vmatrix}$
Expand and collect the quadratic factor.
$D=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

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

2024 JM02