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If $\left|\begin{array}{ccc}\alpha & \beta & \gamma \\ a & b & c \\ l & m & n\end{array}\right|=(-1)^K\left|\begin{array}{ccc}m & n & l \\ b & c & a \\ \beta & \gamma & \alpha\end{array}\right|$, then the least value of $K$ is
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The correct answer is:
3
$\begin{aligned} \left|\begin{array}{lll}\alpha & \beta & \gamma \\ a & b & c \\ l & m & n\end{array}\right| & =(-1)\left|\begin{array}{ccc}\beta & \alpha & \gamma \\ b & a & c \\ m & l & n\end{array}\right|\left(\because C_1 \leftrightarrow C_2\right) \\ & =(-1)^2\left|\begin{array}{lll}\beta & \gamma & \alpha \\ b & c & a \\ m & n & l\end{array}\right|\left(\because C_2 \leftrightarrow C_3\right) \\ & =(-1)^3\left|\begin{array}{ccc}m & n & l \\ b & c & a \\ \beta & \gamma & \alpha\end{array}\right|\left(\because R_1 \leftrightarrow R_3\right) \\ \therefore & K=3\end{aligned}$
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