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$\left | k\times A_{n\times n} \right |=k^{n}\times\left | A_{n\times n} \right |$ (Reference: Determinant when multiplying a matrix by a constant)

$\therefore$ Let the original value of determinant be $\Delta$ i.e. $\Delta=5$

When each element of the matrix is multiplied by $2$, the resultant determinant becomes $40$

$\therefore$ $2^{n}\times 5=40 \Rightarrow 2^{n}=8 \Rightarrow n=3$

Option B is correct.