Question

The value of determinant $$\left|\begin{array}{ccc}a-b& b+c& a\\ b-c& c+a& b\\ c-a& a+b& c\end{array}\right|$$ is

• A. $a^3+b^3+c^3$
• B. $3abc$
• C. $a^3+b^3+c^3-3abc$
• D. $a^3+b^3+c^3+3abc$

Answer 1

The correct option is C $a^3+b^3+c^3-3abc$

$△=\left|\begin{array}{ccc}a-b& b+c& a\\ b-c& c+a& b\\ c-a& a+b& c\end{array}\right|$

Applying $R_1\to R_1+R_2+R_3$, we get
$△=(a+b+c)\left|\begin{array}{ccc}0& 2& 1\\ b-c& c+a& b\\ c-a& a+b& c\end{array}\right|$

Expanding along $R_1$, we get
$△=(a+b+c)[-2c(b-c)+2b(c-a)+(b-c\left)\right(a+b)-(c-a\left)\right(c+a\left)\right]$
$=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$
$=a^3+b^3+c^3-3abc$