Mark the tick against the correct answer in the following: $\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\mathrm{x} & 1 \\ 1 & 1 & 1+\mathrm{y} \end{array}\right|=?$
A. $(x+y)$
B. $(x-y)$
C. $x y$
D. none of these

Solution:

Option(C)
To find: Value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$
We have, $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y\end{array}\right|$
Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$
$\Rightarrow\left|\begin{array}{ccc}
0 & -x & 0 \\
1 & 1+x & 1 \\
1 & 1 & 1+y
\end{array}\right|$
Expanding along $R_{1}$
$\begin{array}{l}
\Rightarrow[x\{(1)(1+y)-(1)(1)\}] \\
\Rightarrow[x\{1+y-1\}] \\
\Rightarrow x y
\end{array}$