I'm trying to answer the following:
Prove the following identity without evaluating the determinants:
a + b c+d e+f a c e b d f
p q r = p q r + p q r
u v w u v w u v w
It comes from the theorem that if A, B and C are n x n matrices who differ in only one row, the ith row, and assume the ith row of C can be obtained by adding corresponding entries in the ith rows of A and B then,
det (C) = det (A) + det (B)
I think I proved it by using co - factor expansion of the determinants on the right hand side of the identity above. I ended up with the following co-factor expansion:
a + b ( qw - rv ) + c + d ( ru - pw ) + e + f ( pv - qu )
And this equals the determinant on the left hand side with a + b being an entry and (qw- rv ) being a minor of that entry etc.
Thus the proof is complete. Or is it ? My query is this: have I just evaluated the determinants without actually proving anything ? My understanding is that the result I showed above will hold for any integer, and therefore the identity is proved for all integers.
Any help would be appreciated.
PS anyone know how I can properly represent determinants on a post ?!?!