Matrix Inverse

The matrix inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that

(1)

where is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation to denote the inverse matrix.

A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is a major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. A matrix possessing an inverse is called an invertible matrix, or nonsingular.

The matrix inverse of a square matrix may be taken in the Wolfram Language using the function Inverse[m].

For a matrix

(2)

the matrix inverse is

			(3)
			(4)

For a matrix

(5)

the matrix inverse is

(6)

A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.

The inverse of a product of matrices and can be expressed in terms of and . Let

(7)

Then

(8)

and

(9)

Therefore,

(10)

so

(11)

where is the identity matrix, and

(12)