Question

If $s=p+q+r$, then value of $\left|\begin{array}{ccc}s+r& p& q\\ r& s+p& q\\ r& p& s+q\end{array}\right|$ is ________.

• A. $s^3$
• B. $2s^3$
• C. $2s^2$
• D. $3s^3$

Answer 1

Answer: (B)

$2s^3$

given $s=p+q+r$

$\left|\begin{array}{ccc}s+r& p& q\\ r& s+p& q\\ r& p& s+q\end{array}\right|$

computing $C_1\to C_1+C_2+C_3$

$=\left|\begin{array}{ccc}s+r+p+q& p& q\\ r+s+p+q& s+p& q\\ r+p+s+q& p& s+q\end{array}\right|$

substituting $s=p+q+r$

$=\left|\begin{array}{ccc}2s& p& q\\ 2s& s+p& q\\ 2s& p& s+q\end{array}\right|$ $=2s\times \left|\begin{array}{ccc}1& p& q\\ 1& s+p& q\\ 1& p& s+q\end{array}\right|$

computing $R_2\to R_2-R_1$ abd $R_3\to R_3-R_1$

$=2s\times \left|\begin{array}{ccc}1& p& q\\ 0& s& 0\\ 0& 0& s\end{array}\right|$

now computing determinant , we get

$=2s\times s^2=2s^3$