Determinant

Definition: Determinant

The determinant of a matrix $A\in F^{n\times n}$ is an element in $F$ which is calculated recursively from the coefficients of $A$:

$$\det (A)=\{\begin{array}{l}{a}_{11},\text{ if }n=1\\ \sum _{i=1}^n(-1{)}^{1+j}{a}_{1j}\det (A_{1j}),\text{ if }n>1\end{array}$$

NOTATION

$$|A|\det (A)$$

Theorem: Distributivity of the Determinant

The determinant is distributive over matrix products:

$$\det (AB)=\det (A)\det (B)=\det (B)\det (A)=\det (BA)$$

PROOF

TODO

Theorem: Determinant of the Transpose

The determinant of the transpose of a $A$ is the same as the determinant of $A$.

$$\det (A^{\mathsf{T}})=\det (A)$$

PROOF

TODO

Theorem: Determinant of the Inverse

If $A$ is invertible, then the determinant of $A^{-1}$ is the reciprocal of $A$‘s determinant:

$$\det (A^{-1})=\frac{1}{\det (A)}$$

PROOF

TODO

Theorem: Determinant of Scalar Multiplication

For the determinant of every square matrix $A\in F^{n\times n}$ and every $\alpha \in F$:

$$\det (\alpha A)={\alpha }^n\det (A)$$

PROOF

TODO