Suppose that $A = (a_{i, j})_{i, j \in [n]}$ is any square matrix. This question is about collecting the known formulas for $\det A$ (which don't depend on $A$ having a special form or dimension) in one place. I suppose I should start with a few:
- The complete expansion says that $\det A = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{k=1}^n a_{k, \sigma(k)}$.
- The Laplace expansion at the $j$-th row is $\det A = \sum_{i=1}^n (-1)^{i+j} a_{i, j} \det(A_{i, j})$, where $A_{i, j}$ is formed from $A$ by deleting the $j$-th column and the $i$-th row (in any order).
- One may form the characteristic polynomial; the determinant will then be the product of its roots (which are, of course, by definition precisely the eigenvalues of $A$ with algebraic multiplicity), or alternatively the constant coefficient times $(-1)^n$.