Determinant

Determinant is a ubiquitous function of a square matrix of size $$n \times n$$, defined as* (p463) $$\text{det}(\mathbf A) = \sum\limits_p \sigma(p) a_{1p_1} a_{2p_2} \dots a_{np_n}$$

where the sum is taken over the $$n!$$ permutations $$p=(p_1,p_2,\dots,p_n)$$ permutations of $$(1,2,\dots,n)$$. Observe that each term $$a_{1p_1} a_{2p_2} \dots a_{np_n}$$ contains exactly one entry from each row and each column of $$\mathbf A$$. The function $$\sigma(p)$$ returns $$+1$$ if the parity in $$p$$ can be restored by an even number of number interchanges, and $$-1$$ if it can be restored by an odd number of interchanges.

Properties of Determinant

 * 1) $$\det(\mathbf{AB})=\det(\mathbf A) \det(\mathbf B)$$ for all $$n\times n$$ matrices
 * 2) $$\det(\matbhf A^{-1}) = 1/\det(\mathbf A)$$ for invertible $$\mathbf A$$

When can determinant be computed easily?
One case is clearly when a matrix $$\mathbf D$$ is diagonal, as $$\det \mathbf D = \prod_i d_{ii}.$$. Also, if the matrix is diagonalizable, Property 1 and 2 above suggest that the determinant can simply be computed

What's the point of determinant?
Determinant directly relates to non-singularity, since $$\mathbf A_{n\times n} \text{ is singular iff } \det(\mathbf A) \neq 0.$$

(But note that