Question

Is the determinant $\Delta=\begin{vmatrix} a-b-c & 2a & 2a \\[0.3em] 2b & b-c-a & 2b \\[0.3em] 2c &2c & c-a-b \end{vmatrix}$

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

Answer 1

Answer: (B)

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