The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix A^(-1) such that

 AA^(-1)=I,
(1)

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

A square matrix A has an inverse iff the determinant |A|!=0 (Lipschutz 1991, p. 45). A matrix possessing an inverse is called nonsingular, or invertible.

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

For a 2×2 matrix

 A=[a b; c d],
(2)

the matrix inverse is

A^(-1)=1/(|A|)[d -b; -c a]
(3)
=1/(ad-bc)[d -b; -c a].
(4)

For a 3×3 matrix

 A=[a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)],
(5)

the matrix inverse is

 A^(-1)=1/(|A|)[|a_(22) a_(23); a_(32) a_(33)| |a_(13) a_(12); a_(33) a_(32)| |a_(12) a_(13); a_(22) a_(23)|;   ; |a_(23) a_(21); a_(33) a_(31)| |a_(11) a_(13); a_(31) a_(33)| |a_(13) a_(11); a_(23) a_(21)|;   ; |a_(21) a_(22); a_(31) a_(32)| |a_(12) a_(11); a_(32) a_(31)| |a_(11) a_(12); a_(21) a_(22)|].
(6)

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

The inverse of a product AB of matrices A and B can be expressed in terms of A^(-1) and B^(-1). Let

 C=AB.
(7)

Then

 B=A^(-1)AB=A^(-1)C
(8)

and

 A=ABB^(-1)=CB^(-1).
(9)

Therefore,

 C=AB=(CB^(-1))(A^(-1)C)=CB^(-1)A^(-1)C,
(10)

so

 CB^(-1)A^(-1)=I,
(11)

where I is the identity matrix, and

 B^(-1)A^(-1)=C^(-1)=(AB)^(-1).
(12)