The determinant is: | a+b b+c c+a |. Adding all three rows to the first row: Row1_new = (a+b)+(b+c)+(c+a) = 2(a+b+c) in each column. So the matrix becomes: | 2(a+b+c) 2(a+b+c) 2(a+b+c) |. Since all elements in the first row are equal, the determinant is 0. (Alternatively, note that the sum of the first and second columns equals twice the third column, making the columns linearly dependent.) Therefore the determinant equals 0.