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$\left|\begin{array}{ccc}
\log e & \log e^2 & \log e^3 \\
\log e^2 & \log e^3 & \log e^4 \\
\log e^3 & \log e^4 & \log e^5
\end{array}\right| \text { is equal to : }$
Options:
\log e & \log e^2 & \log e^3 \\
\log e^2 & \log e^3 & \log e^4 \\
\log e^3 & \log e^4 & \log e^5
\end{array}\right| \text { is equal to : }$
Solution:
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Verified Answer
The correct answer is:
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$\left|\begin{array}{ccc}\log e & \log e^2 & \log e^3 \\ \log e^2 & \log e^3 & \log e^4 \\ \log e^3 & \log e^4 & \log e^5\end{array}\right|$
$=\left|\begin{array}{ccc}\log e & 2 \log e & 3 \log e \\ 2 \log e & 3 \log e & 4 \log e \\ 3 \log e & 4 \log e & 5 \log e\end{array}\right|$
$=\left|\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|$
$=\left|\begin{array}{lll|l}1 & 1 & 1 \\ 2 & 1 & 1 \\ 3 & 1 & 1\end{array}\right| C_2 \rightarrow C_2-C_1$
$=0 \quad(\because$ Two columns are identical $)$
$=\left|\begin{array}{ccc}\log e & 2 \log e & 3 \log e \\ 2 \log e & 3 \log e & 4 \log e \\ 3 \log e & 4 \log e & 5 \log e\end{array}\right|$
$=\left|\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5\end{array}\right|$
$=\left|\begin{array}{lll|l}1 & 1 & 1 \\ 2 & 1 & 1 \\ 3 & 1 & 1\end{array}\right| C_2 \rightarrow C_2-C_1$
$=0 \quad(\because$ Two columns are identical $)$
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