The 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). A matrix possessing
an inverse is called nonsingular,
or invertible.
The matrix inverse of a square matrix may be taken in Mathematica 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)
|