Proof

We prove when \(n=3\).

If we compute the determinant of LHS by expansion across the 2nd row, we have

\begin{eqnarray*}&&\left|\begin{array}{ccc}a_{11}& a_{12}&a_{13}\\ ka_{21}& ka_{22}&ka_{23}\\a_{31}& a_{32}&a_{33}\end{array}\right|\\ && \\ &&=(-1)^{2+1}ka_{21}\left|\begin{array}{cc}a_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|+(-1)^{2+2}ka_{22}\left|\begin{array}{cc}a_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|+(-1)^{2+3}ka_{23}\left|\begin{array}{cc}a_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|\\ && \\&=&k\left((-1)^{2+1}a_{21}\left|\begin{array}{cc}a_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|+(-1)^{2+2}a_{22}\left|\begin{array}{cc}a_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|+(-1)^{2+3}a_{23}\left|\begin{array}{cc}a_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|\right)\end{eqnarray*}

This is equivalent to the determinant of RHS expanded across the 2nd row.

$$\left|\begin{array}{ccc}a_{11}& a_{12}&a_{13}\\ ka_{21}& ka_{22}&ka_{23}\\a_{31}& a_{32}&a_{33}\end{array}\right|=k\left|\begin{array}{ccc}a_{11}& a_{12}&a_{13}\\ a_{21}& a_{22}&a_{23}\\a_{31}& a_{32}&a_{33}\end{array}\right|$$