Check whether (l+m+n) is a factor of the determinant | ccc l+m & … - Vidyadip

Question

Check whether $(l+m+n)$ is a factor of the determinant $\left|\begin{array}{ccc}l+m & m+n & n+l \\ n & l & m \\ 2 & 2 & 2\end{array}\right|$ or not. Give reason.

✓

Answer

Apply $R_{1} \rightarrow R_{1}+R_{2}$
$\left|\begin{array}{ccc}l+m+n & m+n+l & n+l+m \\ n & l & m \\ 2 & 2 & 2\end{array}\right|$
$=2(l+m+n)\left|\begin{array}{ccc}1 & 1 & 1 \\ n & l & m \\ 1 & 1 & 1\end{array}\right|;$ yes $(l+m+n)$ is a factor.

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