# Geometry Formulary

## Inverse of a Matrix
$A^{-1} = \frac{1}{det(A)} Adj(A)$

## Adjugate Matrix
The adjugate of a matrix $A$ is the `transpose` of the `cofactor matrix`:\
$Adj(A) = C^{T}$

### $(i-j)$-minor (AKA $M_{ij}$)
$M_{ij}$ := Determinant of the matrix $B$ got by removing the 
***$i$-row*** and the ***$j$-column*** from matrix $A$

### Cofactors
$C$ is the matrix of `cofactors` of a matrix $A$ where all the elements $c_{ij}$
are defined like this:\
$
c_{ij} = \left( -1\right)^{i + j}M_{ij}
$

## Eigenvalues
By starting from the definition of `eigenvectors`:\
$A\vec{v} = \lambda\vec{v}$

As we can see, the vector $\vec{v}$ was unaffected by this matrix 
multiplication appart from a scaling factor $\lambda$, called `eigenvalue`

By rewriting this formula we get:\
$
\left(A - \lambda I\right)\vec{v} = 0 
$ 

This is solved for:\
$\det(A- \lambda I) = 0$

> [!NOTE]
> If the determinant is 0, then $(A - \lambda I )$ is not invertible, so
> we can't solve the previous equation by using the trivial solution (which
> can't be taken into account since $\vec{v}$ is not $0$  by definition)

## Caley-Hamilton
Each square matrix over a `commutative ring` satisfies its own 
characteristic equation $det(\lambda I - A)$

> [!TIP]
> In other words, once found the characteristic equation, we can 
> substitute the ***unknown*** variable $\lambda$ with the matrix itself
> (***known***),
> powered to the correspondent power