> with(linalg);


> A:=matrix([[a11, a12, a13],\ [b21+c21,b22+c22, b23+c23],\ [a31,a32,a33]]);

A :=	[ a11	a12	a13 ]
	[ b21 + c21	b22 + c22	b23 + c23 ]
	[ a31	a32	a33 ]

You see that the second row of this matrix is the sum of two row vectors. Now let us define matrices B and C:


> B:=matrix([[a11,a12,a13],[b21,b22,b23],[a31,a32,a33]]);

B :=	[ a11	a12	a13 ]
	[ b21	b22	b23 ]
	[ a31	a32	a33 ]

> C:=matrix([[a11,a12,a13],[c21,c22,c23],[a31,a32,a33]]);

C :=	[ a11	a12	a13 ]
	[ c21	c22	c23 ]
	[ a31	a32	a33 ]

Find the determinant of C, A, B:


> da:=det(A); db:=det(B); dc:=det(C);

da := a₁₁ a₃₃ b₂₂ + a₁₁ a₃₃ c₂₂ - a₁₁ a₃₂ b₂₃ - a₁₁ a₃₂ c₂₃ - b₂₁ a₁₂ a₃₃ + b₂₁ a₁₃ a₃₂ - c₂₁ a₁₂ a₃₃ + c₂₁ a₁₃ a₃₂ + a₃₁ a₁₂ b₂₃ + a₃₁ a₁₂ c₂₃ - a₃₁ a₁₃ b₂₂ - a₃₁ a₁₃ c₂₂ db := a₁₁ a₃₃ b₂₂ - a₁₁ a₃₂ b₂₃ - b₂₁ a₁₂ a₃₃ + b₂₁ a₁₃ a₃₂ + a₃₁ a₁₂ b₂₃ - a₃₁ a₁₃ b₂₂ dc := a₁₁ a₃₃ c₂₂ - a₁₁ a₃₂ c₂₃ - c₂₁ a₁₂ a₃₃ + c₂₁ a₁₃ a₃₂ + a₃₁ a₁₂ c₂₃ - a₃₁ a₁₃ c₂₂ You can now see that each term occurring in det(A) or det(B) occurs also in det(A) and vice versa. So indeed det(C)=det(A)+det(B).

Let us ask Maple to check that:

> dc-(da+db);

0