Jacobian Matrix

Given a function of mapping a n-dim input vector x to a m-dim output vector $f : R^{n} \mapsto R^{m}$ , the matrix of all first order partial derivates of this function is the Jacobian matrix J
$J_{ij} = \frac{\partial f _{i}}{\partial x _{j}}$
jacobian matrix $J = \frac{\partial f _{1}}{\partial x _{1}} ⋮ \frac{\partial f _{m}}{\partial x _{1}} \dots ⋱ \dots \frac{\partial f _{1}}{\partial x _{n}} ⋮ \frac{\partial f _{m}}{\partial x _{n}}$

Determinant

The absolute value of the determinant can be thought of as a measure of “how much multiplication by the matrix expands or contracts space”.
Only exists for square matrices
if det(M) = 0, then M is not invertible
The determinant of a 2 × 2 matrix is $a c b d = a d - b c$
determinant of a 3 × 3 matrix is $a d g b e h c f i = a e i + b f g + c d h - ce g - b d i - a f h$
nxn matrix M is $det M = det a_{11} a_{21} ⋮ a_{n 1} a_{12} a_{22} ⋮ a_{n 2} \dots \dots \dots a_{1 n} a_{2 n} ⋮ a_{nn} = \sum_{j_{1} j_{2} \dots j_{n}} (- 1)^{τ (j_{1} j_{2} \dots j_{n})} a_{1 j_{1}} a_{2 j_{2}} \dots a_{n j_{n}}$
where the subscript under the summation j1⁢j2…jn are all permutations of the set {1, 2, …, n}, so there are n! items in total; τ⁡(.) indicates the signature of a permutation.
$d e t (A B) = d e t (A) d e t (B)$