Determinant of a matrix for non-commutative products

Question

Let's suppose we have a $3 \times 3$ matrix $A=(a_{ij})$ whose elements $a_{ij}$ are objects whose product is not commutative: $a_{ij}·a_{kl}\neq a_{kl}·a_{ij}$. Then, which would be the formula for the determinant $|A|$?

Answer 1

In general, for a square matrix of dimension $n$ with coefficients in a ring $\mathcal A$, the determinant is given by:

$$\det(A) = \sum_{\sigma \in \mathfrak S_n} \epsilon(\sigma) \prod_{i=1}^n a_{\sigma(i),i}$$

so you can apply that to the case $n=3$. Now take care that several properties of determinants may not be true anymore if the ring $\mathcal A$ is not commutative! You can have a look at quasideterminant.