Invertible matrix

Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent, that is, for any given matrix they are either all true or all false:

A is invertible, that is, A has an inverse, is nonsingular, or is nondegenerate.

A is row-equivalent to the n-by-n identity matrix I_n.

A is column-equivalent to the n-by-n identity matrix I_n.

A has n pivot positions.

det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring.

A has full rank; that is, rank A = n.

The equation Ax = 0 has only the trivial solution x = 0.

The kernel of A is trivial, that is, it contains only the null vector as an element, ker(A) = {0}.

The equation Ax = b has exactly one solution for each b in Kⁿ.

The columns of A are linearly independent.

The columns of A span Kⁿ.

Col A = Kⁿ.

The columns of A form a basis of Kⁿ.

The linear transformation mapping x to Ax is a bijection from Kⁿ to Kⁿ.

There is an n-by-n matrix B such that AB = I_n = BA.

The transpose A^T is an invertible matrix (hence rows of A are linearly independent, span Kⁿ, and form a basis of Kⁿ).

The number 0 is not an eigenvalue of A.

The matrix A can be expressed as a finite product of elementary matrices.

The matrix A has a left inverse (that is, there exists a B such that BA = I) or a right inverse (that is, there exists a C such that AC = I), in which case both left and right inverses exist and B = C = A⁻¹.

Furthermore, the following properties hold for an invertible matrix A:

• (A⁻¹)⁻¹ = A;
• (kA)⁻¹ = k⁻¹A⁻¹ for nonzero scalar k;
• (Ax)⁺ = x⁺A⁻¹ if A has orthonormal columns, where ⁺ denotes the Moore–Penrose inverse and x is a vector;
• (A^T)⁻¹ = (A⁻¹)^T;
• For any invertible n-by-n matrices A and B, (AB)⁻¹ = B⁻¹A⁻¹. More generally, if A₁, ..., A_k are invertible n-by-n matrices, then (A₁A₂⋅⋅⋅A_k−1A_k)⁻¹ = A⁻¹
  _kA⁻¹
  _k−1⋯A⁻¹
  ₂A⁻¹
  ₁;
• det A⁻¹ = (det A)⁻¹.

The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where we write the rows of V as ${\displaystyle v_{i}^{T}}$ and the columns of U as ${\displaystyle u_{j}}$ for ${\displaystyle 1\leq i,j\leq n}$. Then clearly, the Euclidean inner product of any two ${\displaystyle v_{i}^{T}u_{j}=\delta _{i,j}}$. This property can also be useful in constructing the inverse of a square matrix in some instances where a set of orthogonal vectors (but not necessarily orthonormal vectors) to the columns of U are known and then applying the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse V.

A matrix that is its own inverse, that is, such that A = A⁻¹ and A² = I, is called an involutory matrix.

The adjugate of a matrix ${\displaystyle A}$ can be used to find the inverse of ${\displaystyle A}$ as follows:

If ${\displaystyle A}$ is an ${\displaystyle n\times n}$ invertible matrix, then

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

It follows from the associativity of matrix multiplication that if

${\displaystyle \mathbf {AB} =\mathbf {I} \ }$

for finite square matrices A and B, then also

${\displaystyle \mathbf {BA} =\mathbf {I} \ }$[1]

Over the field of real numbers, the set of singular n-by-n matrices, considered as a subset of R^n×n, is a null set, that is, has Lebesgue measure zero. This is true because singular matrices are the roots of the determinant function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory, almost all n-by-n matrices are invertible.

Furthermore, the n-by-n invertible matrices are a dense open set in the topological space of all n-by-n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n-by-n matrices.

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; such matrices are said to be ill-conditioned.

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 upper and lower triangular matrices, which are easier to invert.

A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it is convenient to find a suitable starting seed:

${\displaystyle X_{k+1}=2X_{k}-X_{k}AX_{k}.}$

Victor Pan and John Reif have done work that includes ways of generating a starting seed.[2][3] Byte magazine summarised one of their approaches.[4]

Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be the already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to the Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic.

The Cayley–Hamilton theorem allows the inverse of ${\displaystyle A}$ to be expressed in terms of det(${\displaystyle A}$), traces and powers of ${\displaystyle A}$:[5]

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots ,k_{n-1}}\prod _{l=1}^{n-1}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} (\mathbf {A} ^{l})^{k_{l}},}$

where ${\displaystyle n}$ is dimension of ${\displaystyle A}$, and ${\displaystyle \operatorname {tr} (A)}$ is the trace of matrix ${\displaystyle A}$ given by the sum of the main diagonal. The sum is taken over ${\displaystyle s}$ and the sets of all ${\displaystyle k_{l}\geq 0}$ satisfying the linear Diophantine equation

${\displaystyle s+\sum _{l=1}^{n-1}lk_{l}=n-1.}$

The formula can be rewritten in terms of complete Bell polynomials of arguments ${\displaystyle t_{l}=-(l-1)!\operatorname {tr} (A^{l})}$ as

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\sum _{s=1}^{n}\mathbf {A} ^{s-1}{\frac {(-1)^{n-1}}{(n-s)!}}B_{n-s}(t_{1},t_{2},\ldots ,t_{n-s}).}$

If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A is invertible and its inverse is given by

${\displaystyle \mathbf {A} ^{-1}=\mathbf {Q} \mathbf {\Lambda } ^{-1}\mathbf {Q} ^{-1},}$

where ${\displaystyle \mathbf {Q} }$ is the square (N×N) matrix whose i-th column is the eigenvector ${\displaystyle q_{i}}$ of ${\displaystyle \mathbf {A} }$, and ${\displaystyle \mathbf {\Lambda } }$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, that is, ${\displaystyle \Lambda _{ii}=\lambda _{i}}$. Since ${\displaystyle \mathbf {Q} }$ is formed from the eigenvectors of ${\displaystyle \mathbf {A} }$, it is guaranteed to be an orthogonal matrix, therefore ${\displaystyle \mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }}$. Furthermore, because ${\displaystyle \mathbf {\Lambda } }$ is a diagonal matrix, its inverse is easy to calculate:

${\displaystyle \left[\Lambda ^{-1}\right]_{ii}={\frac {1}{\lambda _{i}}}.}$

If matrix A is positive definite, then its inverse can be obtained as

${\displaystyle \mathbf {A} ^{-1}=(\mathbf {L} ^{*})^{-1}\mathbf {L} ^{-1},}$

where L is the lower triangular Cholesky decomposition of A, and L* denotes the conjugate transpose of L.

Writing the transpose of the matrix of cofactors, known as an adjugate matrix, can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors:

${\displaystyle \mathbf {A} ^{-1}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\mathbf {C} ^{\mathrm {T} }={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}{\begin{pmatrix}\mathbf {C} _{11}&\mathbf {C} _{21}&\cdots &\mathbf {C} _{n1}\\\mathbf {C} _{12}&\mathbf {C} _{22}&\cdots &\mathbf {C} _{n2}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {C} _{1n}&\mathbf {C} _{2n}&\cdots &\mathbf {C} _{nn}\\\end{pmatrix}}}$

so that

${\displaystyle \left(\mathbf {A} ^{-1}\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} ^{\mathrm {T} }\right)_{ij}={1 \over {\begin{vmatrix}\mathbf {A} \end{vmatrix}}}\left(\mathbf {C} _{ji}\right)}$

where |A| is the determinant of A, C is the matrix of cofactors, and C^T represents the matrix transpose.

The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows:[6]

${\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b\\c&d\\\end{bmatrix}}^{-1}={\frac {1}{\det \mathbf {A} }}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}={\frac {1}{ad-bc}}{\begin{bmatrix}\,\,\,d&\!\!-b\\-c&\,a\\\end{bmatrix}}.}$

This is possible because 1/(ad − bc) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes.

The Cayley–Hamilton method gives

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det \mathbf {A} }}\left[\left(\operatorname {tr} \mathbf {A} \right)\mathbf {I} -\mathbf {A} \right].}$

A computationally efficient 3 × 3 matrix inversion is given by

${\displaystyle \mathbf {A} ^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,B&\,C\\\,D&\,E&\,F\\\,G&\,H&\,I\\\end{bmatrix}}^{\mathrm {T} }={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}\,A&\,D&\,G\\\,B&\,E&\,H\\\,C&\,F&\,I\\\end{bmatrix}}}$

(where the scalar A is not to be confused with the matrix A). If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by

${\displaystyle {\begin{alignedat}{6}A&={}&(ei-fh),&\quad &D&={}&-(bi-ch),&\quad &G&={}&(bf-ce),\\B&={}&-(di-fg),&\quad &E&={}&(ai-cg),&\quad &H&={}&-(af-cd),\\C&={}&(dh-eg),&\quad &F&={}&-(ah-bg),&\quad &I&={}&(ae-bd).\\\end{alignedat}}}$

The determinant of A can be computed by applying the rule of Sarrus as follows:

${\displaystyle \det(\mathbf {A} )=aA+bB+cC.}$

The Cayley–Hamilton decomposition gives

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left[{\frac {1}{2}}\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)\mathbf {I} -\mathbf {A} \operatorname {tr} \mathbf {A} +\mathbf {A} ^{2}\right].}$

The general 3 × 3 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix ${\displaystyle \mathbf {A} =\left[\mathbf {x} _{0},\;\mathbf {x} _{1},\;\mathbf {x} _{2}\right]}$ (consisting of three column vectors, ${\displaystyle \mathbf {x} _{0}}$, ${\displaystyle \mathbf {x} _{1}}$, and ${\displaystyle \mathbf {x} _{2}}$) is invertible, its inverse is given by

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}{\begin{bmatrix}{(\mathbf {x_{1}} \times \mathbf {x_{2}} )}^{\mathrm {T} }\\{(\mathbf {x_{2}} \times \mathbf {x_{0}} )}^{\mathrm {T} }\\{(\mathbf {x_{0}} \times \mathbf {x_{1}} )}^{\mathrm {T} }\end{bmatrix}}.}$

The determinant of A, ${\displaystyle \det(\mathbf {A} )}$, is equal to the triple product of ${\displaystyle \mathbf {x_{0}} }$, ${\displaystyle \mathbf {x_{1}} }$, and ${\displaystyle \mathbf {x_{2}} }$—the volume of the parallelepiped formed by the rows or columns:

${\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}$

The correctness of the formula can be checked by using cross- and triple-product properties and by noting that for groups, left and right inverses always coincide. Intuitively, because of the cross products, each row of ${\displaystyle \mathbf {A} ^{-1}}$ is orthogonal to the non-corresponding two columns of ${\displaystyle \mathbf {A} }$ (causing the off-diagonal terms of ${\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} }$ be zero). Dividing by

${\displaystyle \det(\mathbf {A} )=\mathbf {x} _{0}\cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}$

causes the diagonal elements of ${\displaystyle \mathbf {I} =\mathbf {A} ^{-1}\mathbf {A} }$ to be unity. For example, the first diagonal is:

${\displaystyle 1={\frac {1}{\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2})}}\mathbf {x_{0}} \cdot (\mathbf {x} _{1}\times \mathbf {x} _{2}).}$

With increasing dimension, expressions for the inverse of A get complicated. For n = 4, the Cayley–Hamilton method leads to an expression that is still tractable:

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\left[{\frac {1}{6}}\left((\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} \mathbf {A} ^{2}+2\operatorname {tr} \mathbf {A} ^{3}\right)\mathbf {I} -{\frac {1}{2}}\mathbf {A} \left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}\operatorname {tr} \mathbf {A} -\mathbf {A} ^{3}\right].}$

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

${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&-\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\\-(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\end{bmatrix}},}$

(1)

where A, B, C and D are matrix sub-blocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and D − CA⁻¹B must be nonsingular.[7]) This strategy is particularly advantageous if A is diagonal and D − CA⁻¹B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion.

This technique was reinvented several times and is due to Hans Boltz (1923), who used it for the inversion of geodetic matrices, and Tadeusz Banachiewicz (1937), who generalized it and proved its correctness.

The nullity theorem says that the nullity of A equals the nullity of the sub-block in the lower right of the inverse matrix, and that the nullity of B equals the nullity of the sub-block in the upper right of the inverse matrix.

The inversion procedure that led to Equation (1) performed matrix block operations that operated on C and D first. Instead, if A and B are operated on first, and provided D and A − BD⁻¹C are nonsingular,[8] the result is

${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}&-(\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}\\-\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}&\quad \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}\end{bmatrix}}.}$

(2)

Equating Equations (1) and (2) leads to

${\displaystyle (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}=\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}\,}$

(3)

${\displaystyle (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}=\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\,}$

${\displaystyle \mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}=(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}\,}$

${\displaystyle \mathbf {D} ^{-1}+\mathbf {D} ^{-1}\mathbf {C} (\mathbf {A} -\mathbf {BD} ^{-1}\mathbf {C} )^{-1}\mathbf {BD} ^{-1}=(\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\,}$

where Equation (3) is the Woodbury matrix identity, which is equivalent to the binomial inverse theorem.

Since a blockwise inversion of an n × n matrix requires inversion of two half-sized matrices and 6 multiplications between two half-sized matrices, it can be shown that a divide and conquer algorithm that uses blockwise inversion to invert a matrix runs with the same time complexity as the matrix multiplication algorithm that is used internally.[9] There exist matrix multiplication algorithms with a complexity of O(n^2.3727) operations, while the best proven lower bound is Ω(n² log n).[10]

This formula simplifies significantly when the upper right block matrix ${\displaystyle B}$ is the zero matrix. This formulation is useful when the matrices ${\displaystyle A}$ and ${\displaystyle D}$ have relatively simple inverse formulas (or pseudo inverses in the case where the blocks are not all square. In this special case, the block matrix inversion formula stated in full generality above becomes

${\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {0} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}&\mathbf {0} \\-\mathbf {D} ^{-1}\mathbf {CA} ^{-1}&\mathbf {D} ^{-1}\end{bmatrix}}.}$

If a matrix A has the property that

${\displaystyle \lim _{n\to \infty }(\mathbf {I} -\mathbf {A} )^{n}=0}$

then A is nonsingular and its inverse may be expressed by a Neumann series:[11]

${\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }(\mathbf {I} -\mathbf {A} )^{n}.}$

Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner. Note that a truncated series can be accelerated exponentially by noting that the Neumann series is a geometric sum. As such, it satisfies

${\displaystyle \sum _{n=0}^{2^{L}-1}(\mathbf {I} -\mathbf {A} )^{n}=\prod _{l=0}^{L-1}(\mathbf {I} +(\mathbf {I} -\mathbf {A} )^{2^{l}})}$.

Therefore, only ${\displaystyle 2L-2}$ matrix multiplications are needed to compute ${\displaystyle 2^{L}}$ terms of the sum.

More generally, if A is "near" the invertible matrix X in the sense that

${\displaystyle \lim _{n\to \infty }(\mathbf {I} -\mathbf {X} ^{-1}\mathbf {A} )^{n}=0\mathrm {~~or~~} \lim _{n\to \infty }(\mathbf {I} -\mathbf {A} \mathbf {X} ^{-1})^{n}=0}$

then A is nonsingular and its inverse is

${\displaystyle \mathbf {A} ^{-1}=\sum _{n=0}^{\infty }\left(\mathbf {X} ^{-1}(\mathbf {X} -\mathbf {A} )\right)^{n}\mathbf {X} ^{-1}~.}$

If it is also the case that A − X has rank 1 then this simplifies to

${\displaystyle \mathbf {A} ^{-1}=\mathbf {X} ^{-1}-{\frac {\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} )\mathbf {X} ^{-1}}{1+\operatorname {tr} (\mathbf {X} ^{-1}(\mathbf {A} -\mathbf {X} ))}}~.}$

If A is a matrix with integer or rational coefficients and we seek a solution in arbitrary-precision rationals, then a p-adic approximation method converges to an exact solution in ${\displaystyle O(n^{4}\log ^{2}n)}$, assuming standard ${\displaystyle O(n^{3})}$ matrix multiplication is used.[12] The method relies on solving n linear systems via Dixon's method of p-adic approximation (each in ${\displaystyle O(n^{3}\log ^{2}n)}$) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.[13]

Given an ${\displaystyle n\times n}$ square matrix ${\displaystyle \mathbf {X} =[x^{ij}]}$, ${\displaystyle 1\leq i,j\leq n}$, with ${\displaystyle n}$ rows interpreted as ${\displaystyle n}$ vectors ${\displaystyle \mathbf {x} _{i}=x^{ij}\mathbf {e} _{j}}$ (Einstein summation assumed) where the ${\displaystyle \mathbf {e} _{j}}$ are a standard orthonormal basis of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (${\displaystyle \mathbf {e} _{i}=\mathbf {e} ^{i},\mathbf {e} _{i}\cdot \mathbf {e} ^{j}=\delta _{i}^{j}}$), then using Clifford algebra (or Geometric Algebra) we compute the reciprocal (sometimes called dual) column vectors ${\displaystyle \mathbf {x} ^{i}=x_{ji}\mathbf {e} ^{j}=(-1)^{i-1}(\mathbf {x} _{1}\wedge \cdots \wedge ()_{i}\wedge \cdots \wedge \mathbf {x} _{n})\cdot (\mathbf {x} _{1}\wedge \ \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})^{-1}}$ as the columns of the inverse matrix ${\displaystyle \mathbf {X} ^{-1}=[x_{ji}]}$. Note that, the place "${\displaystyle ()_{i}}$" indicates that "${\displaystyle \mathbf {x} _{i}}$" is removed from that place in the above expression for ${\displaystyle \mathbf {x} ^{i}}$. We then have ${\displaystyle \mathbf {X} \mathbf {X} ^{-1}=[\mathbf {x} _{i}\cdot \mathbf {x} ^{j}]=[\delta _{i}^{j}]=\mathbf {I} _{n}}$, where ${\displaystyle \delta _{i}^{j}}$ is the Kronecker delta. We also have ${\displaystyle \mathbf {X} ^{-1}\mathbf {X} =[(\mathbf {e} _{i}\cdot \mathbf {x} ^{k})(\mathbf {e} ^{j}\cdot \mathbf {x} _{k})]=[\mathbf {e} _{i}\cdot \mathbf {e} ^{j}]=[\delta _{i}^{j}]=\mathbf {I} _{n}}$, as required. If the vectors ${\displaystyle \mathbf {x} _{i}}$ are not linearly independent, then ${\displaystyle (\mathbf {x} _{1}\wedge \mathbf {x} _{2}\wedge \cdots \wedge \mathbf {x} _{n})=0}$ and the matrix ${\displaystyle \mathbf {X} }$ is not invertible (has no inverse).

Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy, found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.[15]

Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace-to-world object transformations, and physical simulations.

Matrix inversion also plays a significant role in the MIMO (Multiple-Input, Multiple-Output) technology in wireless communications. The MIMO system consists of N transmit and M receive antennas. Unique signals, occupying the same frequency band, are sent via N transmit antennas and are received via M receive antennas. The signal arriving at each receive antenna will be a linear combination of the N transmitted signals forming an N × M transmission matrix H. It is crucial for the matrix H to be invertible for the receiver to be able to figure out the transmitted information.