Proof

We prove when \(n=3\).

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

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

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

Thus we obtain

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