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# determinants watch

1. 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

any solutions?
2. 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).

Solution:
Spoiler:
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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.

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Updated: January 11, 2015
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