Find the value of the determinant

Question:

Find the value of the determinant $\left|\begin{array}{ccc}243 & 156 & 300 \\ 81 & 52 & 100 \\ -3 & 0 & 4\end{array}\right|$.

Solution:

$\begin{array}{ccc}\mid 243 & 156 & 300 \\ 81 & 52 & 100 \\ -3 & 0 & 4 \mid\end{array}$

$=\mid 243-(81 \times 3) 156-(52 \times 3) 300-(100 \times 3) 81$ $\begin{array}{llll}52 & 100-3 & 0 & 4 \mid\end{array}$

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

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

$\begin{array}{lll}81 & 52 & 100\end{array}$

$\begin{array}{lll}-3 & 0 & 4 \mid\end{array}$

$=0$

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