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प्रश्न
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
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उत्तर
\[ ∆ = \begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
\[ = \begin{vmatrix}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 9 & 16 & 25 & 36 \\ 16 & 25 & 36 & 49\end{vmatrix}\]
\[ = \begin{vmatrix}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 5 & 7 & 9 & 11 \\ 7 & 9 & 11 & 13\end{vmatrix} \left[ \text{ Applying } R_3 \to R_3 - R_2 and R_4 \to R_4 - R_3 \right]\]
\[ = \begin{vmatrix}1 & 4 & 9 & 16 \\ 4 & 9 & 16 & 25 \\ 7 & 9 & 11 & 13 \\ 7 & 9 & 11 & 13\end{vmatrix} = 0 \left[\text{ Applying }R_3 \to 2 + R_3 \right]\]
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