#algebra 

The definition of a determinant (определитель/детерминант) applies only to square [[Matrix|matrices]].

Written as $\LARGE \det(A)$ or $\LARGE |A|$ (NOT MODULUS)
## Determinant of a $\LARGE 2 \times 2$ matrix
![[Pasted image 20240920215512.png]]

## Determinant of a $\LARGE 3 \times 3$ matrix
Sarrus' rule (способ/правило Саррюса):
![[Pasted image 20250102215900.png]]
![[Pasted image 20250102215916.png]]

## Determinant of a $\LARGE n \times n$ matrix
Let $\LARGE A$ be an $\LARGE n \times n$ matrix:
1) The **minor** $\LARGE M_{ij}$ of the element $\LARGE a_{ij}$ is the determinant of the matrix obtained by deleting the ith row and jth column of A
2) The **cofactor** $\LARGE A_{ij}$ of the element $\LARGE a_{ij}$ is $\LARGE A_{ij}=(-1)^{i+j}M_{ij}$

>[!Example of operations with minors and cofactors]-
>![[Pasted image 20240920220256.png]]

Then:

$$\LARGE \det(A)=a_{r1}A_{r1}+a_{r2}A_{r2}+...+a_{rn}A_{rn}$$
sum of each element of row $\LARGE r$ multiplied by its cofactor.
OR
you can go along any row/column

This method is called **Laplace expansion**


## Use
If the determinant of a matrix is 0, it doesn't have an [[Inverse matrix|inverse]]. If it is NOT 0, it does have an inverse.

## Transformation property
![[Pasted image 20240920230633.png]]
![[Pasted image 20240920230645.png]]

## Other properties
$$\LARGE \det{AB}=\det{A}\cdot\det{B}$$
(proof using mathematical induction)