# Linear Algebra/Determinant

The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these properties:

• It is linear on the rows of the matrix.
${\displaystyle \det {\begin{bmatrix}\ddots &\vdots &\ldots \\\lambda a_{1}+\mu b_{1}&\cdots &\lambda a_{n}+\mu b_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}=\lambda \det {\begin{bmatrix}\ddots &\vdots &\cdots \\a_{1}&\cdots &a_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}+\mu \det {\begin{bmatrix}\ddots &\vdots &\cdots \\b_{1}&\cdots &b_{n}\\\cdots &\vdots &\ddots \end{bmatrix}}}$
• If the matrix has two equal rows its determinant is zero.
• The determinant of the identity matrix is 1.

It is possible to prove that ${\displaystyle \det A=\det A^{T}}$, making the definition of the determinant on the rows equal to the one on the columns.

## Properties

• The determinant is zero if and only if the rows are linearly dependent.
• Changing two rows changes the sign of the determinant:
${\displaystyle \det {\begin{bmatrix}\cdots \\{\mbox{row A}}\\\cdots \\{\mbox{row B}}\\\cdots \end{bmatrix}}=-\det {\begin{bmatrix}\cdots \\{\mbox{row B}}\\\cdots \\{\mbox{row A}}\\\cdots \end{bmatrix}}}$

• The determinant is a multiplicative map in the sense that
${\displaystyle \det(AB)=\det(A)\det(B)\,}$ for all n-by-n matrices ${\displaystyle A}$ and ${\displaystyle B}$.

This is generalized by the Cauchy-Binet formula to products of non-square matrices.

• It is easy to see that ${\displaystyle \det(rI_{n})=r^{n}\,}$ and thus
${\displaystyle \det(rA)=\det(rI_{n}\cdot A)=r^{n}\det(A)\,}$ for all ${\displaystyle n}$-by-${\displaystyle n}$ matrices ${\displaystyle A}$ and all scalars ${\displaystyle r}$.
• A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have
${\displaystyle \det(A^{-1})=\det(A)^{-1}.\,}$

Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.

A matrix and its transpose have the same determinant:

${\displaystyle \det(A^{\top })=\det(A).\,}$

The determinants of a complex matrix and of its conjugate transpose are conjugate:

${\displaystyle \det(A^{*})=\det(A)^{*}.\,}$

## Theorems

### Existance

Using Laplace's formula for the determinant

### Binet's theorem

${\displaystyle \det(AB)=\det A\cdot \det B}$