Question

Is this a legitimate proof of the product of determinants property in tensor notation?

Answer 1

I'll start with the matrix C which is the product of the matrices A and B.
\[c^i_k = a^i_jb^j_k\]The determinant of C is \[\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t \]
by the definition of multiplication plugging in and rearranging:
\[\frac{1}{3!}\delta_{ijk}^{rst} a^i_lb^l_ra^j_mb^m_sa^k_nb^n_t = \frac{1}{3!}\delta_{ijk}^{rst} a^i_la^j_ma^k_nb^l_rb^m_sb^n_t \]
now the step I'm unsure of is that I know I can turn this into this on its own:
\[\delta_{ijk}^{rst} = \frac{1}{3!} \delta_{ijk}^{lmn} \delta_{lmn}^{rst} \]
But can I plug this into the equation and mix the indices like that so I get: \[ \frac{1}{3!}\frac{1}{3!} \delta_{ijk}^{lmn} \delta_{lmn}^{rst} a^i_la^j_ma^k_nb^l_rb^m_sb^n_t \] which is the end of the proof since these are the determinants of A and B multiplied together.