# 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.