See [[Exterior algebra of vector space#^bb3af2|functoriality of exterior powers]]. Determinants play an important role in the theory of volume and integration (Sheldon, p.213). Notably, they appear in the [[Change of variables formula|change of variables formula]]. # Definitions ###### Axiomatic definition For a square [[Matrices|matrix]] $A\in \mathbb{F}^{n\times n},$ where $\mathbb{F}$ is a field or even a commutative ring, there is an associated scalar $\det A$ called the **determinant** of $A$ which, regarded as a function, $\begin{align} \mathbb{F}^{n\times n} &\to \mathbb{F} \\ A &\mapsto \det A \end{align}$has the following properties with respect to [[Elementary row operations are equivalent to pre-multiplying by elementary matrices|elementary matrices]]: 1. $\det(S_{ij}A)=-\det A$, i.e. the determinant is alternating; 2. $\det(M_{j}(\lambda)A)=\lambda \det A$, i.e. linearity. 3. $\det(A_{ij}(\lambda)A)=\det A$ 4. $\det I_{n}=1$. Like this we see how [[Row reduction algorithm|Gaussian elimination]] lends itself to computing the determinant. TBC: uniqueness. ###### Leibniz formula See [[Leibniz formula for determinants]]: $\det A = \sum_{\sigma \in S_{n}} \text{sgn}(\sigma)a_{1 \sigma_{1}} a_{2 \sigma_{2}} \dots a_{n \sigma(n)}$where $S_{n}$ denotes the symmetric group of degree $n.$ ###### Cofactor (Laplace) expansion See [[Cofactor expansion of determinant]]. ###### In terms of characteristic polynomial/ eigenvalues Sheldon defines the determinant as $(-1)^{n}$ times the constant term of the [[Characteristic polynomial of linear operator|characteristic polynomial]] which to me seems cyclic since the CP of $A$ itself is usually defined as $\det(A-\lambda I_{n})$. # Properties The determinant $\det A$ of a matrix $A$ satisfies the following properties: 1. If a row of $A$ is zero, then $\det A=0.$ 2. If two rows of $A$ are identical, then $\det A=0.$ 3. If $A$ is a diagonal matrix, that is $A_{ij}=0$ whenever $i\neq j$, then $\det A=a_{11}a_{22}\cdots a_{nn}.$ 4. If $A$ is an upper triangular matrix, that is $A_{ij}=0$ whenever $i>j$ then $\det A=$ $a_{11}a_{22}\cdots a_{nn}.$ 5. If $A$ is a lower triangular matrix, that is $A_{ij}=0$ whenever $i<j$, then $\det A=$ $a_{11}a_{22}\cdots a_{nn}.$ # External Links See https://ncatlab.org/nlab/show/determinant.