# Inverses

## Inverses

## Basics of Matrix Inverses

- An
**inverse matrix**is a square matrix that, when multiplied with the original matrix, results in the identity matrix. - Not all square matrices have inverses. A matrix that does not have an inverse is said to be
**singular**or**non-invertible**. - If a matrix has an inverse, it is a
**nonsingular**or**invertible**matrix. - A matrix multiplied by its inverse always gives the
**identity matrix**. That is, if`A`

is the original matrix and`A^-1`

is its inverse, then`A*A^-1`

and`A^-1*A`

both give the identity matrix.

## Calculating Inverse Matrices

- The inverse of a
**2x2 matrix**,`A`

, with elements`a=(a,b;c,d)`

can be found using the formula`A^-1 = 1/(ad-bc)*(d,-b;-c,a)`

. Note that`ad - bc`

is the**determinant**of the matrix. - For matrices larger than 2x2, the inverse can be calculated through various methods such as
**Gaussian elimination**or by using the**adjugate matrix**and the determinant of the original matrix. - The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix.

## Properties of Inverse Matrices

- The inverse of a matrix, if it exists, is unique.
- The inverse of the inverse of a matrix is the original matrix itself. That is,
`(A^-1)^-1 = A`

. - The inverse of the product of two matrices equals the product of their inverses in reverse order. That is,
`(AB)^-1 = B^-1*A^-1`

. - If a matrix is singular, i.e., its determinant is 0, it does not have an inverse.

## Applications of Inverse Matrices

- Inverse matrices are used extensively in the solution of systems of
**linear equations**. - They are crucial in fields such as
**linear regression**, where the inverse of a covariance matrix is used to estimate coefficients. - They also have significant applications in
**physics and engineering**, for example, calculating electrical networks or solving differential equations.