matrices, help please?

Question

Let A be an nxn matrix:
Prove that:
If B is the matrix that results when a multiple of one row of A is added to another row, then det(B) = det(A).
I know that I should use signed elementary products, but i have no clue how to do this. Help me, please! thank you!

Answer 1

Say a[i,j] are the elements of A. The definition of the determinant is:

detA = Sum over P: (-1)^P * a[1,P(1)] * a[2,P(2)]*...* a[n,P(n)]

where the sum is over all permutations P of (1,2,...,n) and (-1)^P is -1,+1 for odd and even permutations respectively. (P(i) is the image of i under P).

Now let's say (for example) that matrix B is created multiplying the second row by a constant c and adding it to the first row:

a[1,i] --> a[1,i]+c*a[2,i]

Then the determinant of this new matrix will be:

detB = Sum over P:
(-1)^P * (a[1,P(1)]+c*a[2,P(1)]) * a[2,P(2)]*...* a[n,P(n)]

= det A +
c * Sum over P: (-1)^P * a[2,P(1)] * a[2,P(2)]*...* a[n,P(n)]

but the last sum is zero, because for each permutation P there is exactly one other permutation differing only by a transposition T of (1,2)

P' = PT , P = P'T

and P and P' have opposite parity so the two terms generated by each of these n!/2 pairs cancel each other --> detB = detA