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Matrix Inverse

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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). The so-called invertible matrix theorem is 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 nonsingular, or invertible.

The matrix inverse of a square matrix m may be taken in the Wolfram Language 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)

See also

Drazin Inverse, Gauss-Jordan Elimination, Gaussian Elimination, LU Decomposition, Matrix, Matrix 1-Inverse, Matrix Addition, Matrix Multiplication, Moore-Penrose Matrix Inverse, Nonsingular Matrix, Pseudoinverse, Singular Matrix, Strassen Formulas Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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References

Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 11, 1962.Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications. New York: Wiley, 1977.Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989.Jodár, L.; Law, A. G.; Rezazadeh, A.; Watson, J. H.; and Wu, G. "Computations for the Moore-Penrose and Other Generalized Inverses." Congress. Numer. 80, 57-64, 1991.Lipschutz, S. "Invertible Matrices." Schaum's Outline of Theory and Problems of Linear Algebra, 2nd ed. New York: McGraw-Hill, pp. 44-45, 1991.Nash, J. C. Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, 2nd ed. Bristol, England: Adam Hilger, pp. 24-26, 1990.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Is Matrix Inversion an N^3 Process?" §2.11 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 95-98, 1992.Rosser, J. B. "A Method of Computing Exact Inverses of Matrices with Integer Coefficients." J. Res. Nat. Bur. Standards Sect. B. 49, 349-358, 1952.

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Matrix Inverse

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Matrix Inverse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixInverse.html

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