i cant do second part
i tried adding a new row and collumn to C with 0 entries for ith entry where i>n+1 and tried to found the determinant of that
- Thread Starter
- 10-01-2015 21:33
- 11-01-2015 02:48
HINT: Consider the Laplace expansions, and notice that since all other elements of the matrix are zero apart from the parts of and there is a certain invariance (Subbing out).
Suppose is the determinant of the matrix , where the th column of the original matrix is replaced by the new column of , except for . So the original matrix has a principal diagonal of .
Then consider the Laplace expansion of your new matrix with an matrix up top.
Then . Where are the cofactors of .
Now by the inductive hypothesis , as is always an matrix.
Now take as your new matrix, then you have that , by the Laplace expansion.
Now considering the Laplace expansion of , it is clear that , thus .
In fact there's a really easy way to prove the fact that the determinant of is indeed if you know the idea behind Leibniz's formula for determinants, but I guess it doesn't really work for an induction problem like this one.Last edited by 0x2a; 11-01-2015 at 02:53.