#algebra 

Let A be a square $\LARGE n \times n$ [[Matrix|matrix]]. An **inverse matrix** $\LARGE A^{-1}$ is a matrix such that:
$\LARGE AA^{-1}=A^{-1}A=I_n$, where $\LARGE I_n$ is an [[Identity matrix|identity matrix]].

If a matrix has an inverse, it is called **invertible**.

**(MUST BE A SQUARE MATRIX!!)**
## Finding an inverse matrix

1. Suppose you have a **square** $\LARGE n \times n$ matrix
2. Append an $\LARGE n \times n$ identity matrix to the right of the given matrix so that you get an $\LARGE n \times 2n$ matrix
3. Using [[Gaussian elimination]], transform the original half of the matrix into an identity matrix
4. The right half of the matrix is an inverse to the given

>[!Example]-
>![[Pasted image 20240920205827.png]]
>![[Pasted image 20240920205901.png]]

A matrix that doesn't have an inverse is called **singular matrix**:
![[Pasted image 20240920210001.png]]

## Finding an inverse matrix using [[Determinant of a matrix|determinants]]
To find an inverse matrix of matrix $\LARGE A$ using determinants:

1) verify that $\LARGE \det{A}\neq0$
2) replace each $\LARGE a_{ij}$ with its [[Determinant of a matrix|cofactor]]
3) find the [[Transpose of a matrix|transpose of this matrix]]
4) multiply it by $\LARGE \frac{1}{\det{A}}$.
5) the result is the matrix $\LARGE A^{-1}$
![[Pasted image 20241016181645.png]]