In linear algebra, an n-by-n (square) matrix $A$ is called invertible, non-singular, or regular if there exists an n-by-n matrix $B$ such that

$AB\; =\; BA\; =\; I\_n\; \backslash$

where $I\_n$ denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix $B$ is uniquely determined by $A$ and is called the inverse of $A$, denoted by $A^\{-1\}$. It follows from the theory of matrices that if

$AB\; =\; I\; \backslash$

for square matrices $A$ and $B$, then also

$BA\; =\; I\; \backslash$.

A square matrix that is not invertible is called singular or degenerate. While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring.

As a rule of thumb, almost all matrices are invertible. Over the field of real numbers, this can be made precise as follows: the set of singular n-by-n matrices, considered as a subset of $R^\{n\; \backslash times\; n\}$, is a null set, i.e., has Lebesgue measure zero. Intuitively, this means that if you pick a random square matrix over the reals, the probability that it will be singular is zero. This is true because singular matrices can be thought of as the roots of the polynomial function given by the determinant. In practice however, one may encounter non-invertible matrices. And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic.

Matrix inversion is the process of finding the matrix $B$ that satisfies the prior equation for a given invertible matrix $A$.

Properties of invertible matrices

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Let $A$ be a square n by n matrix over a field $K$ (for example the field $R$ of real numbers). Then the following statements are equivalent:

• $A$ is invertible.
• $A$ is row-equivalent to the n-by-n identity matrix $I\_n$.
• $A$ has n pivot positions.
• det $A$ ≠ 0.
• rank $A$ = n.
• The equation $Ax\; =\; 0$ has only the trivial solution $x\; =\; 0$ (i.e. Null $A\; =\; 0$).
• The equation $Ax\; =\; b$ has exactly one solution for each $b$ in $K^n$.
• The columns of $A$ are linearly independent.
• The columns of $A$ span $K^n$ (i.e. Col $A\; =\; K^n$).
• The columns of $A$ form a basis of $K^n$.
• The linear transformation mapping $x$ to $Ax$ is a bijection from $K^n$ to $K^n$.
• There is an n by n matrix $B$ such that $AB\; =\; I\_n$.
• The transpose $A^T$ is an invertible matrix.
• The matrix times its transpose, $A^T\; \backslash times\; A$ is an invertible matrix.
• The number 0 is not an eigenvalue of $A$.

In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.

The inverse of an invertible matrix $A$ is itself invertible, with

$\backslash left(A^\{-1\}\backslash right)^\{-1\}\; =\; A$.

The inverse of an invertible matrix $A$ multiplied by a scalar $k$ yields the product of the inverse of both the matrix and the scalar

$\backslash left(kA\backslash right)^\{-1\}\; =\; k^\{-1\}A^\{-1\}$.

The product of two invertible matrices $A$ and $B$ of the same size is again invertible, with the inverse given by

$\backslash left(AB\backslash right)^\{-1\}\; =\; B^\{-1\}A^\{-1\}$

(note that the order of the factors is reversed.) As a consequence, the set of invertible n-by-n matrices forms a group, known as the general linear group Gl(n).

Proof for matrix product rule

If $A\_1$, $A\_2$, ..., $A\_n$ are nonsingular square matrices over a field, then

$(A\_1A\_2\backslash cdots\; A\_n)^\{-1\}\; =\; A\_n^\{-1\}A\_\{n-1\}^\{-1\}\backslash cdots\; A\_1^\{-1\}$

It becomes evident why this is the case if one attempts to find an inverse for the product of the $A\_i$s from first principles, that is, that we wish to determine $B$ such that

$(A\_1A\_2\backslash cdots\; A\_n)B=I$

where $B$ is the inverse matrix of the product. To remove $A\_1$ from the product, we can then write

$A\_1^\{-1\}(A\_1A\_2\backslash cdots\; A\_n)B=A\_1^\{-1\}I$

which would reduce the equation to

$(A\_2A\_3\backslash cdots\; A\_n)B=A\_1^\{-1\}I$

Likewise, then, from

$A\_2^\{-1\}(A\_2A\_3\backslash cdots\; A\_n)B=A\_2^\{-1\}A\_1^\{-1\}I$

which simplifies to

$(A\_3A\_4\backslash cdots\; A\_n)B=A\_2^\{-1\}A\_1^\{-1\}I$

If one repeat the process up to $A\_n$, the equation becomes

$B=A\_n^\{-1\}A\_\{n-1\}^\{-1\}\backslash cdots\; A\_2^\{-1\}A\_1^\{-1\}I$

$B=A\_n^\{-1\}A\_\{n-1\}^\{-1\}\backslash cdots\; A\_2^\{-1\}A\_1^\{-1\}$

but $B$ is the inverse matrix, i.e $B\; =\; (A\_1A\_2\backslash cdots\; A\_n)^\{-1\}$ so the property is established

Methods of matrix inversion

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Gauss-Jordan elimination

Gauss-Jordan elimination is an algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the LU decomposition which generates an upper and a lower triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating mn-by-mn matrices as m-by-m matrices of n-by-n matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices).

Analytic solution

Writing another special matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices (since this method is essentially recursive, it becomes inefficient for large matrices). To determine the inverse, we calculate a matrix of cofactors:

$A^\{-1\}=\{1\; \backslash over\; \backslash begin\{vmatrix\}A\backslash end\{vmatrix\}\}\backslash left(C\_\{ij\}\backslash right)^\{T\}=\{1\; \backslash over\; \backslash begin\{vmatrix\}A\backslash end\{vmatrix\}\}$

\begin{pmatrix} C_{11} & C_{21} & \cdots & C_{j1} \\ C_{12} & \ddots & & C_{j2} \\ \vdots & & \ddots & \vdots \\ C_{1i} & \cdots & \cdots & C_{ji} \\ \end{pmatrix}

where |A| is the determinant of A, C_ij is the matrix cofactor, and A^T represents the matrix transpose.

In most practical applications, it is not necessary to invert a matrix to solve a system of linear equations, however it is necessary that the matrix involved is invertible.

Decomposition techniques like LU decomposition, are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.

Inversion of 2 x 2 matrices

The cofactor equation listed above yields the following result for 2 x 2 matrices. Inversion of these matrices can be done easily as follows:

$A^\{-1\}\; =\; \backslash begin\{bmatrix\}$

a & b \\ c & d \\ \end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix}

Inversion of 3 x 3 matrices

The cofactor equation listed above yields the following result for 3 x 3 matrices. Inversion of these matrices can be done quite easily as follows:

$A^\{-1\}\; =\; \backslash begin\{bmatrix\}$

a & b & c\\ d & e & f \\ g & h & i \\ \end{bmatrix}^{-1} = \frac{1}{|A|} \begin{bmatrix} ei - fh & ch - bi & bf - ce \\ fg - di & ai - cg & cd - af \\ dh - eg & bg - ah & ae - bd \end{bmatrix}

$|A|\; =\; a(ei-fh)\; -\; b(di-fg)\; +\; c(dh-eg)\; \backslash$

Blockwise inversion

Matrices can also be inverted blockwisely by using the following analytic inversion formula:

where A, B, C and D are matrix sub-blocks of arbitrary size. This strategy is particularly advantageous if A is diagonal and $(D-CA^\{-1\}B)$ (the Schur complement of $A$) is a small matrix, since they are the only matrices requiring to be inverted.

This technique was invented by Volker Strassen, who also invented the Strassen algorithm for fast(er) matrix multiplication.

The derivative of the matrix inverse

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Suppose that the matrix A depends on a parameter t. Then the derivative of the inverse of A with respect to t is given by

$\backslash frac\{\backslash mathrm\{d\}A^\{-1\}\}\{\backslash mathrm\{d\}t\}\; =\; -\; A^\{-1\}\; \backslash frac\{\backslash mathrm\{d\}A\}\{\backslash mathrm\{d\}t\}\; A^\{-1\}.$

This formula can be found by differentiating the identity

$A^\{-1\}A\; =\; I$.

The Moore-Penrose pseudoinverse

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Some of the properties of inverse matrices are shared by (Moore-Penrose) pseudoinverses, which can be defined for any m-by-n matrix.

See also

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• matrix decomposition
• matrix multiplication
• pseudoinverse (Moore-Penrose inverse)
• singular value decomposition

References

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• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 28.4: Inverting matrices, pp.755–760.

External links

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• Inverse Matrix Calculator
• Moore Penrose Pseudoinverse

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