Inverse Matrix · aiwiki.page

An inverse matrix is a matrix that reverses another matrix under multiplication. In linear algebra, the inverse of a square matrix $A$ is denoted $A^{-1}$ and satisfies

AA^{-1}=A^{-1}A=I,

where $I$ is the identity matrix of the same size. A matrix possessing an inverse is called invertible or nonsingular; one without an inverse is singular. Inversion reverses the transformation represented by a matrix, rather than taking the reciprocal of each individual entry. (math.mit.edu)

Definition and existence

The usual theory concerns $n\times n$ matrices with entries in a field, particularly the real numbers or complex numbers. The inverse, when it exists, is unique. Indeed, if $B$ and $C$ are both inverses of $A$ , associativity gives

B=B(AC)=(BA)C=C.

For square matrices over a field, a one-sided inverse is sufficient: $AB=I$ implies $BA=I$ . The finite-dimensional, square-matrix assumption is essential to this statement. (arxiv.org)

Several equivalent conditions characterize invertibility:

The determinant satisfies $\det(A)\ne0$ .
The rank of $A$ is $n$ .
Its columns are linearly independent and form a basis of the coordinate space.
Its null space contains only the zero vector.
The linear system $Ax=b$ has exactly one solution for every $b$ .
Elimination produces $n$ nonzero pivots, allowing row exchanges. (math.mit.edu)

Geometrically, $A$ represents a linear map on a finite-dimensional vector space. Invertibility means that this map is an isomorphism: every output corresponds to exactly one input. A singular map loses information by sending some nonzero vector to zero, making complete reversal impossible. (math.ucla.edu)

Formulas and examples

For a $2\times2$ matrix,

A=\begin{pmatrix}a&b\\c&d\end{pmatrix},

the inverse exists precisely when $ad-bc\ne0$ , and then

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

For example,

A=\begin{pmatrix}2&1\\5&3\end{pmatrix}, \qquad A^{-1}=\begin{pmatrix}3&-1\\-5&2\end{pmatrix}.

Direct multiplication in either order yields $I$ . By contrast, the matrix with rows $(1,2)$ and $(2,4)$ is singular because its determinant is zero. (math.mit.edu)

For an arbitrary invertible square matrix, the exact formula

A^{-1}=\frac{\operatorname{adj}(A)}{\det(A)}

uses the adjugate matrix, the transpose of the matrix of signed minors. This identity connects inversion with determinants and gives explicit algebraic expressions for the inverse entries. Computing many minors, however, makes it an unattractive general-purpose procedure for large matrices. (math.ucla.edu)

Algebraic properties

For invertible matrices $A$ and $B$ of the same size,

(A^{-1})^{-1}=A,\qquad (AB)^{-1}=B^{-1}A^{-1}.

The reversed order is important: reversing two successive transformations requires undoing the last transformation first. It cannot generally be replaced by $A^{-1}B^{-1}$ , because matrix multiplication is not commutative. (math.mit.edu)

Inversion also interacts with the transpose and determinant:

(A^T)^{-1}=(A^{-1})^T, \qquad \det(A^{-1})=\frac{1}{\det(A)}.

A real orthogonal matrix therefore has the especially simple inverse $A^{-1}=A^T$ . (math.mit.edu)

If $Av=\lambda v$ with $v\ne0$ , invertibility ensures $\lambda\ne0$ , and multiplication by $A^{-1}$ gives

A^{-1}v=\lambda^{-1}v.

Thus the eigenvalues of the inverse are reciprocals of those of $A$ , with the corresponding eigenvectors unchanged. (linear.pugetsound.edu)

Computation and numerical accuracy

A standard exact method is Gauss–Jordan elimination, a variant of Gaussian elimination. Elementary row operations transform the augmented matrix

[A\mid I]\longrightarrow[I\mid A^{-1}].

The same operations that reduce $A$ to the identity accumulate its inverse in the right-hand block. Failure to obtain a pivot in every column indicates singularity. Equivalently, the columns of $A^{-1}$ are the solutions of $Ax_j=e_j$ , where $e_j$ are the standard basis vectors. (math.mit.edu)

In numerical linear algebra, inversion can be organized through LU decomposition with pivoting and subsequent triangular solves. For solving $Ax=b$ , explicitly forming $A^{-1}$ is usually unnecessary; numerical libraries instead solve the system using a factorization. This avoids computing an entire inverse when only its action on particular right-hand sides is required. (netlib.org)

Mathematical invertibility does not guarantee accurate results in floating-point arithmetic. Sensitivity is measured by the condition number

\kappa_2(A)=\frac{\sigma_{\max}(A)}{\sigma_{\min}(A)},

where the $\sigma$ 's are singular values. A large ratio indicates ill-conditioning: small perturbations or rounding errors can produce substantial inaccuracies. An inversion routine may return a result for an ill-conditioned matrix without reporting failure. (numpy.org)

Rectangular matrices and generalized inverses

A rectangular matrix cannot have a two-sided inverse of the ordinary kind. Nevertheless, an $m\times n$ matrix with full column rank has a left inverse, while one with full row rank has a right inverse. These one-sided inverses need not be unique. (ocw.mit.edu)

The Moore–Penrose pseudoinverse, denoted $A^+$ , extends inversion to rectangular and singular matrices. It can be constructed using singular value decomposition by reciprocating nonzero singular values and leaving zero singular values at zero. For invertible square matrices, $A^+=A^{-1}$ ; otherwise, $A^+b$ is the minimum-Euclidean-norm solution among those minimizing $\|Ax-b\|_2$ . This provides a generalized solution even when an exact solution is absent or nonunique. (numpy.org)