if  ∇ =\(\begin{bmatrix} a_{11} & a_{12} & a_{13} \\[0.3em] a_{21} & a_{22} & a_{23} \\[0.3em] a_{31} & a_{32} & a_{33} \end{bmatrix}\) and A_ij is co- factors of a_ij.

⇒ ∇= a₁₁\(\begin{bmatrix} a_{22} & a_{23} \\[0.3em] a_{32} & a_{33} \end{bmatrix}\) _- a₁₂\(\begin{bmatrix} a_{21} & a_{23} \\[0.3em] a_{31} & a_{33} \end{bmatrix}\)₊ a₁₃\(\begin{bmatrix} a_{21} & a_{22} \\[0.3em] a_{31} & a_{32} \end{bmatrix}\)(By expanding determinant along row R₁)

⇒ ∇= a₁₁(-1)¹⁺¹\(\begin{bmatrix} a_{22} & a_{23} \\[0.3em] a_{32} & a_{33} \end{bmatrix}\) - a₁₂(−1) ¹⁺²\(\begin{bmatrix} a_{21} & a_{23} \\[0.3em] a_{31} & a_{33} \end{bmatrix}\)+ a₁₃(−1)¹⁺¹ \(\begin{bmatrix} a_{21} & a_{22} \\[0.3em] a_{31} & a_{32} \end{bmatrix}\).

⇒ ∇= a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃. (By definition of cofactor of element of matrix. )

Also, if we expanding determinant along row R₂, we get ∇= a₂₁A₂₁ + a₂₂A₂₂ + a₂₃A₂₃. 

And if we expanding determinant along row R₃, we get ∇= a₃₁A₃₁ + a₃₂A₃₂ + a₃₃A₃₃.

Therefore, we can write ∇ as ∇= a_i1A_i1 + a_i2A_i2 + a_i3A_i3, where 1 ≤ i≤ 3.

Therefore, ∇ = \(\sum_{j}^{3}\) a_ijA_ij , where 1 ≤  i ≤ 3.