Question

Is the determinant $\Delta =\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$

• A. a perfect square
• B. a perfect cube
• C. zero
• D. none of these.

Answer 1

Answer: (B)

Operate $R_1+R_2+R_3$
$\Delta =\left|\begin{array}{ccc}a+b+c& a+b+c& a+b+c\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$
= $(a+b+c)\left|\begin{array}{ccc}1& 1& 1\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$
Operate $C_3-C_1;C_2-C_1$
= $(a+b+c)\left|\begin{array}{ccc}1& 0& 0\\ 2b& -(a+b+c)& 0\\ 2c& 0& -(a+b+c)\end{array}\right|$
= $(a+b+c{)}^3$