Determinants, Adding and Scaling Rows

Adding and Scaling Rows

Scaling a row or column scales the determinant. Prove this by using det₂. If the j^th row, or the j^th column, is multiplied by k, then exactly one element in every permutation product is multiplied by k, each product is multiplied by k, and the determinant is multiplied by k.

If a row or column is 0 the determinant is 0.

If row j is the sum of two vectors, the determinant of the matrix containing that row is the sum of the determinants of the two matrices that use those two vectors. Again, look at a permutation product. The j^th entry in such a product is x+y, and we can separate it into two products, one using x and one using y. Regroup terms to get the sum of two determinants.

Assume two rows are proportional, i.e. one row is k times another. If a permutation product uses column 7 from the first row and column 9 from the second, there is another permutation with 7 and 9 swapped. One product has kxy times some stuff from other rows, and the other product has xky times the same stuff. The products are identical, except for the parity of the permutations, which is reversed when two elements are swapped. Therefore the products are opposite, and cancel each other out. This leaves a determinant of 0.

Now for something counterintuitive. We can subtract a multiple of one row from another and it won't change the determinant at all. To illustrate, subtract k times the second row from the first. The determinant of this new matrix is the original determinant minus the determinant produced by replacing the first row with k times the second. But now the first and second rows are proportional, so the latter determinant is 0. The new determinant is the same as the original.