> 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