Question

Find the value of the determinant 
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]

 

Answer 1

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

\[ \Rightarrow \Delta = \begin{vmatrix} 101 & 102 & 103\\104 & 105 & 106 \\107 & 108 & 109 \end{vmatrix} = 0\]