Find the value of the determinant

Solution:

Let $\Delta=\mid \begin{array}{lll}101 & 102 & 103\end{array}$

$\begin{array}{lll}104 & 105 & 106\end{array}$

$\begin{array}{lll}107 & 108 & 109\end{array}$

$\Delta=\mid \begin{array}{lll}101 & 1 & 2\end{array}$

$\begin{array}{lll}104 & 1 & 2\end{array}$

$\begin{array}{lll}107 & 1 & 2\end{array}$

$\left[\right.$ Applying $C_{2} \rightarrow C_{2}-C_{1}$ and $\left.C_{3} \rightarrow C_{3}-C_{1}\right]$

$=2 \mid \begin{array}{lll}101 & 1 & 1\end{array}$

$\begin{array}{lll}104 & 1 & 1\end{array}$

$\begin{array}{lll}107 & 1 & 1\end{array}$

$=0$

Since two columns are identitical, the value of the determinant is zero.

$\Rightarrow \Delta=\mid \begin{array}{lll}101 & 102 & 103\end{array}$

$\begin{array}{lll}104 & 105 & 106\end{array}$

$\begin{array}{lll}107 & 108 & 109\end{array} \mid=0$