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Question
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
Solution
\[\text{ Let }\Delta = \begin{vmatrix} 101 & 102 & 103\\104 & 105 & 106 \\107 & 108 & 109 \end{vmatrix}\]
\[ \Delta = \begin{vmatrix} 101 & 1 & 2\\104 & 1 & 2\\107 & 1 & 2 \end{vmatrix} \left[\text{ Applying }C_2 \to C_2 - C_1\text{ and }C_3 \to C_3 - C_1 \right]\]
\[ = 2\begin{vmatrix} 101 & 1 & 1\\104 & 1 & 1\\107 & 1 & 1 \end{vmatrix}\]
\[ = 0 \]
Since two columns are identitical, the value of the determinant is zero .
\[ \Rightarrow \Delta = \begin{vmatrix} 101 & 102 & 103\\104 & 105 & 106 \\107 & 108 & 109 \end{vmatrix} = 0\]
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