Question

Find the value of the following determinants.
$\left|\begin{array}{ccc}a& b& c\\ a-b& b-c& c-a\\ b+c& c+a& a+b\end{array}\right|$

• A. ${\left(a+b+c\right)}^2$
• B. $a^3+b^3+c^3-3abc$
• C. ${\left(a+b+c\right)}^3$
• D. $a^3+b^3+c^3$

Answer 1

Answer: (B)

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

Applying $C_1\to C_1+C_2+C_3$, we get

$\Delta =\left|\begin{array}{ccc}a+b+c& b& c\\ 0& b-c& c-a\\ 2\left(a+b+c\right)& a+c& a+b\end{array}\right|$

Taking $\left(a+b+c\right)$ common from $C_1$, we get

$\Delta =\left(a+b+c\right)\left|\begin{array}{ccc}1& b& c\\ 0& b-c& c-a\\ 2& a+c& a+b\end{array}\right|$

Applying $R_3\to R_3-2R_1$, we get

$\Delta =\left(a+b+c\right)\left|\begin{array}{ccc}1& b& c\\ 0& b-c& c-a\\ 0& a+c-2b& a+b-2c\end{array}\right|$

Expanding along $C_1$, we get

$\Delta =\left(a+b+c\right)\left[\left(b-c\right)\left(a+b-2c\right)-\left(c-a\right)\left(a+c-2b\right)\right]$

$=\left(a+b+c\right)\left[a^2+b^2+c^2-ab-bc-ac\right]$

$=a^3+b^3+c^3-3abc$