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प्रश्न
Evaluate the following determinant:
\[\begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}\]
उत्तर
\[ ∆ = \begin{vmatrix}1 & 3 & 9 & 27 \\ 3 & 9 & 27 & 1 \\ 9 & 27 & 1 & 3 \\ 27 & 1 & 3 & 9\end{vmatrix}\]
\[ = 1\begin{vmatrix}9 & 27 & 1 \\ 27 & 1 & 3 \\ 1 & 3 & 9\end{vmatrix} - 3\begin{vmatrix}3 & 27 & 1 \\ 9 & 1 & 3 \\ 27 & 3 & 9\end{vmatrix} + 9\begin{vmatrix}3 & 9 & 1 \\ 9 & 27 & 3 \\ 27 & 1 & 9\end{vmatrix} - 27\begin{vmatrix}3 & 9 & 27 \\ 9 & 27 & 1 \\ 27 & 1 & 3\end{vmatrix}\]
\[ = 1\left\{ 9(9 - 9) - 27(243 - 3) + 1(81 - 1) \right\} - 3\left\{ 3(9 - 9) - 27(81 - 81) + 1(27 - 27) \right\} + 9\left\{ 3(243 - 3) - 9(81 - 81) + 1(9 - 729) \right\} - 27\left\{ (81 - 1) - 9(27 - 27) + 27(9 - 729) \right\}\]
\[ = 1\left\{ 0 - 6480 + 80 \right\} - 3\left\{ 0 - 0 + 0 \right\} + 9\left\{ 720 - 0 - 720 \right\} - 27\left\{ 80 - 0 - 19440 \right\}\]
\[ = - 6400 + 522720\]
\[ = 516320\]
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