## Geometric Perspective: The determinant is the signed change in volume of a unit space by some linear transformation $\det(A)= \pm \frac{\text{Volume}(Ax|x \in S)}{\text{Volume}(S)}$ where the sign is negative when the image gets flipped by A. This determinant ratio is constant regardless of $S$. In the 2D case: $\det\begin{bmatrix}a & b \\ c & d\end{bmatrix} = ad-bc$ An $n$ x $n$ matrix $A$ is invertible if and only if $\det(A)\neq 0$ Properties of determinants: - for an $n$ x $n$ matrix $A$, $\det(\lambda A) = \lambda^{n}A$ - for $n$ x $n$ matrices $A$ and $B$, $\det(AB) = \det(A)\det(B)$ - for any invertible matrix $A$, $\det(A^{-1}) = \frac{1}{\det A}$ ## Algebraic Perspective: $\det(A) = (-1)^{\text{\# of row swaps}} \cdot (\text{product of pivots in ref}(A))$ Row swaps flip the image, therefore need to be accounted for ![[Pasted image 20251229172421.png|400]]