# Determinant

The determinant is actually super widely used for a bunch of things that I originally did not expect, which is why I dedicated its own separate page.

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix.

What actually is the determinant?

Iβve learned it so many times, but Iβve only really learned to apply formulas. Never got a fundamental understanding of how it works.

### Matrix Inverse

The inverse is of $A$ is equal to $A_{β1}=detA1βadjA$ Thus, $A$ is invertible if and only if $detAξ=0$

### For $2Γ2$ Matrix

We can just apply a simple formula for determinants and adjugates.
Let
$A=[acβbdβ]βM_{2Γ2}(R)$
The *determinant* of $A$ is
$detA=adβbc$
and the *adjugate* of $A$ is
$adjA=[dβcββbaβ]$

See below for the more generalized definitions.

### For $nΓn$ matrix

#### Cofactor

We need to use the Cofactor expansion to calculate the determinant. Letβs first define cofactor.

Let $AβM_{nΓn}(R)$ and let $A(i,j)$ be the $(nβ1)Γ(nβ1)$ matrix obtained from $A$ by deleting the *i*th row and *j*th column of $A$. The (i, j)-*cofactor* of $A$, denoted by $Cij$ , is
$C_{ij}=(β1)_{i+j}detA(i,j)$
so the cofactor is just a number.

#### Determinant

Let $AβM_{nΓn}(R)$. For any $i=1,β¦,n$, we define the determinant of $A$ as the cofactor expansion of $A$ along ANY row or column $i$ of $A$. $detA=a_{i1}C_{i1}+a_{i2}C_{i2}+β―+a_{in}C_{in}$

#### Cofactor Matrix and Adjugate

The cofactor matrix of $A$ is $cofA=[C_{ij}]βM_{nΓn}(R)$

The adjugate of $A$ is $adjA=[C_{ij}]_{T}βM_{nΓn}(R)$

### Properties of determinants

Let $A,BβM_{nΓn}(R)$ and $kβR$. $det(kA)=k_{n}detA$ $det(A_{k})=(detA)_{k}$ $det(AB)=(detA)(detB)=(detB)(detA)=det(BA)$ $det(A_{T})=det(A)$

Danger

$det(A+B)ξ=det(A)+det(B)$

### Determinants and Area

Somehow the determinant is related to the area? Also volume??

### Determinants and Volume

Very similar derivation as area.