Levi-Civita and Kronecker delta identity, proof with determinants

[Pifagor]

Homework Statement

I'm trying to understand a proof of the LC-KD identity involving determinants (see attachment), from the book Introduction to Tensor Calculus and Continuum Mechanics by Herinbockel.
What is the author saying in the last line of text? How can we sum the deltas in the upper right corner, shouldn't we sum the three determinants as a whole, since we're doing that on the other side of the equation? And how does the sum 3 come into the picture anyway?

Homework Equations

See attachment.

The Attempt at a Solution

It seems I should write out the whole determinant for all i,j,k,r,s,t, but that would not make the proof any easier than doing it by "brute force" in the first place!

Thanks for any help.

 

[tiny-tim]

Welcome to PF!

Hi Pifagor! Welcome to PF!

See attachment.

erm …

no attachment! ​

 

[Pifagor]

.../four letter word /... now it should be OK

 

[tiny-tim]

that's better!

if you use the δⁱ_t on the top right to replace all the is by ts in its cofactor, you get minus the cofactor for δⁱ_i

the same for δⁱ_s, so you have (3 - 1 - 1) times that cofactor