Question:
$\left|\begin{array}{ccc}x+4 & x & x \\ x & x+4 & x \\ x & x & x+4\end{array}\right|$
Solution:
Given, $\left|\begin{array}{ccc}x+4 & x & x \\ x & x+4 & x \\ x & x & x+4\end{array}\right|$
$=\left|\begin{array}{ccc}3 x+4 & 3 x+4 & 3 x+4 \\ x & x+4 & x \\ x & x & x+4\end{array}\right|$ [Applying $\left.R_{1} \rightarrow R_{1}+R_{2}+R_{3}\right]$
$=(3 x+4)\left|\begin{array}{ccc}1 & 1 & 1 \\ x & x+4 & x \\ x & x & x+4\end{array}\right|$
$=(3 x+4)\left|\begin{array}{ccc}0 & 0 & 1 \\ -4 & 4 & x \\ 0 & -4 & x+4\end{array}\right|=16(3 x+4)$
[Applying $C_{1} \rightarrow C_{1}-C_{2}, C_{2} \rightarrow C_{2}-C_{3}$ ]
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