# Intuitive explanation for the general determinant formula?

by jjepsuomi
Tags: determinant, linear algebra
 P: 1,622 Often the determinant is just defined by the formula: $$\det(a_{ij}) = \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma)a_{1\sigma(1)} \cdots a_{n\sigma(n)}$$ On the other hand, if you define the determinant as the unique alternating multilinear functional $\det:\mathbb{R}^n \times \cdots \times \mathbb{R}^n \rightarrow \mathbb{R}$ (where the product is taken n-times) satisfying $\det(I) = 1$, then you can recover the formula above for the determinant. Edit: I suppose this is not really an intuitive explanation, but hopefully it helps a little.
 Mentor P: 18,333 A nice and intuitive explanation of the determinant is that it just represents a signed volume. For example, given the vectors (a,b) and (c,d) in $\mathbb{R}^2$. Then we can look at the parallelogram formed by (0,0), (a,b), (c,d) and (a+c,b+d). The area of this parallellogram is given by the absolute value of $$det \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)$$ Of course the determinant has a sign as well. This is why we call the determinant the signed volume. That is: if we exchange (a,b) and (c,d), then we get the opposite area. The sign is useful for determining orientation.