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Question
The value of \[\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}\]
Options
52
0
513
59
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Solution
\[\begin{vmatrix}5^2 & 5^3 & 5^4 \\ 5^3 & 5^4 & 5^5 \\ 5^4 & 5^5 & 5^6\end{vmatrix}\]
\[ = 5^2 \times 5^3 \times 5^4 \begin{vmatrix} 1 & 5 & 5^2 \\1 & 5 & 5^2 \\1 & 5 & 5^2 \end{vmatrix} \left[\text{ Taking out common factors from }R_{1,} R_2 , R_3 \right]\]
\[ = 5^2 \times 5^3 \times 5^4 \times 5 \begin{vmatrix} 1 & 1 & 5^2 \\1 & 1 & 5^2 \\1 & 1 & 5^2 \end{vmatrix}\]
\[ = 5^2 \times 5^3 \times 5^4 \times 0\]
\[ = 0\]
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