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.

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.

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.

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.