Question
Using the factor theorem it is found that $$b + c, c + a$$ and $$a + b$$ are three factors of the determinant \[\left| {\begin{array}{*{20}{c}}
{ - 2a}&{a + b}&{a + c} \\
{b + a}&{ - 2b}&{b + c} \\
{c + a}&{c + b}&{ - 2c}
\end{array}} \right|.\] The other factor in the value of the determinant is
A.
$$4$$
B.
$$2$$
C.
$$a + b + c$$
D.
None of these
Answer :
$$4$$
Solution :
As the determinant is of the third degree in $$a, b, c,$$ we get
\[\left| {\begin{array}{*{20}{c}}
{ - 2a}&{a + b}&{a + c} \\
{b + a}&{ - 2b}&{b + c} \\
{c + a}&{c + b}&{ - 2c}
\end{array}} \right| = \lambda \left( {b + c} \right)\left( {c + a} \right)\left( {a + b} \right),\]
where $${\lambda}$$ is independent of $$a, b, c.$$
Putting $$a = 0, b = 1, c = 2,$$
\[\left| {\begin{array}{*{20}{c}}
0&1&2 \\
1&{ - 2}&3 \\
2&3&{ - 4}
\end{array}} \right| = \lambda \cdot 3 \cdot 2 \cdot 1.\,\,\,\,\,{\text{Now find }}\lambda .\]