Question

Find the value of the determinant \[\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .\]

Answer 1

\[\begin{vmatrix} 243 & 156 & 300\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix}\] 
\[ = \begin{vmatrix} 243 - \left( 81 \times 3 \right) & 156 - \left( 52 \times 3 \right) & 300 - \left( 100 \times 3 \right)\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix} \left[\text{ Applying }R_1 \to R_1 - 3 R_2 \right]\] 
\[ = \begin{vmatrix} 0 & 0 & 0\\ 81 & 52 & 100\\ - 3 & 0 & 4 \end{vmatrix} \] 
\[ = 0\]