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I'm trying to learn the basics of Tensor calculus using a free online book (Introduction to Tensor Calculus and Continuum Mechanics), and I got stuck on this question (Part 2 in book, after non-math introduction).

link to part 2, questions (mine #19) at end:

http://www.math.odu.edu/~jhh/part2.PDF" [Broken]

1. The problem statement, all variables and given/known data

Let A and B denote 3x3 matrices with elements [tex]A_{ij}[/tex] and [tex]B_{ij}[/tex]

respectively. Show that if C = AB is a matrix product, then det(C) = det(A)*det(B)

where det = determinant.

Hint: use the result from example 1.1-9

2. Relevant equations

det(A) = [tex]e_{ijk}A_{1i}A_{2j}A_{3k}[/tex]

3. The attempt at a solution

The matrix multiplication of A*B =C, in indical notation, is

[tex]C_{ij}=A_{im}B_{mj}[/tex] (I think) where the first subscript in A,B and C is the row number and the second subscript the column of the matrix.

Then, plugging into 'relevant equation' above, we get

det(C) = [tex]e_{ijk}C_{1i}C_{2j}C_{3k}[/tex]

det(C) = [tex]e_{ijk}(A_{1m}B_{mi})(A_{2n}B_{nj})(A_{3x}B_{xk})[/tex]

Then, I compare this to just multiplying det(A)*det(B)

det(A) * det(B) = [tex](e_{ijk}A_{1i}A_{2j}A_{3k})(e_{rst}B_{1r}B_{2s}B_{3t})[/tex]

However, from here, I can't seem to make the connection between the two

I would try to expand it, but I don't think this is the way to do it, since it would get rid of the advantage of indical notation. Help would be appreciated! Thxs!

PS: on a side note, whenever you preview a post, the template is pasted again to the end of the post, which is annoying. Maybe someone can fix this...

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# Tensor notation-Determinants

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