Invertible matrix

 ( Linear Algebra ) 

In linear algebra, an invertible matrix ( non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it can be multiplied by another matrix to yield the identity matrix. Invertible matrices are the same size as their inverse.

The inverse of a matrix represents the inverse operation, meaning if a matrix is applied to a particular vector, followed by applying the matrix's inverse, the result is the original vector.

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Definition An -by- square matrix is called invertible if there exists an -by- square matrix such that$$\backslash mathbf\{AB\}\; =\; \backslash mathbf\{BA\}\; =\; \backslash mathbf\{I\}\_n\; ,$$where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication.

Sheldon Axler (2014). 9783319110790, Springer Publishing. ISBN 9783319110790

If this is the case, then the matrix is uniquely determined by , and is called the inverse of , denoted by . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix.

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Examples Consider the following 2-by-2 matrix:

The matrix $\backslash mathbf\{A\}$ is invertible, as it has inverse which can be confirmed by computing

To check that it is invertible without finding an inverse, $\backslash det\; \backslash mathbf\{A\}\; =\; -\backslash frac\{1\}\{2\}$ can be computed, which is non-zero.

On the other hand, this is a non-invertible matrix:

We can see the rank of this 2-by-2 matrix is 1, which is , so it is non-invertible. Additionally, we can compute that the determinant of $\backslash mathbf\{C\}$ is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.

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Methods of matrix inversion

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Gaussian elimination Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the identity matrix. Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix.

For example, take the following matrix:

The first step to compute its inverse is to create the augmented matrix

1 & -1 & 0 & 1
\end{array}\!\!\right) 
     

Call the first row of this matrix $R\_1$ and the second row $R\_2$. Then, add row 1 to row 2 $(R\_1\; +\; R\_2\; \backslash to\; R\_2).$ This yields

0 & \tfrac{1}{2} & 1 & 1
     

\end{array}\!\!\right)

Next, subtract row 2, multiplied by 3, from row 1 $(R\_1\; -\; 3\backslash ,\; R\_2\; \backslash to\; R\_1),$ which yields

0 & \tfrac{1}{2} & 1 & 1
     

\end{array}\!\!\right)

Finally, multiply row 1 by −1 $(-R\_1\; \backslash to\; R\_1)$ and row 2 by 2 $(2\backslash ,\; R\_2\; \backslash to\; R\_2).$ This yields the identity matrix on the left side and the inverse matrix on the right:

Thus, It works because the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices ($\backslash mathbf\; E\_n$), such as $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\{n-1\}\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; A\; =\; \backslash mathbf\; I$

Applying right-multiplication using $\backslash mathbf\; A^\{-1\},$ we get $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\{n-1\}\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; I\; =\; \backslash mathbf\; I\; \backslash mathbf\; A^\{-1\}.$ And the right side $\backslash mathbf\; I\; \backslash mathbf\; A^\{-1\}\; =\; \backslash mathbf\; A^\{-1\},$ which is the inverse we want.

To obtain $\backslash mathbf\; E\_n\; \backslash mathbf\; E\_\{n-1\}\; \backslash cdots\; \backslash mathbf\; E\_2\; \backslash mathbf\; E\_1\; \backslash mathbf\; I,$ we create the augmented matrix by combining with and applying Gaussian elimination. The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes , the right portion applied the same elementary row operation sequence will become .

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Newton's method 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:

$X\_\{k+1\}\; =\; 2X\_k\; -\; X\_k\; A\; X\_k$

Victor Pan and John Reif have done work that includes ways of generating a starting seed.

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. That 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 from round-off error.

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Cayley–Hamilton method The Cayley–Hamilton theorem allows the inverse of to be expressed in terms of , traces and powers of :A proof can be found in the Appendix B of

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash frac\{1\}\{\backslash det(\backslash mathbf\{A\})\}\; \backslash sum\_\{s=0\}^\{n-1\}\; \backslash mathbf\{A\}^s\; \backslash sum\_\{k\_1,k\_2,\backslash ldots,k\_\{n-1\}\}\; \backslash prod\_\{l=1\}^\{n-1\}\; \backslash frac\{(-1)^\{k\_l\; +\; 1\}\}\{l^\{k\_l\}k\_l!\}\; \backslash operatorname\{tr\}\backslash left(\backslash mathbf\{A\}^l\backslash right)^\{k\_l\},$

where is size of , and is the trace of matrix given by the sum of the main diagonal. The sum is taken over and the sets of all $k\_l\; \backslash geq\; 0$ satisfying the linear Diophantine equation

$s\; +\; \backslash sum\_\{l=1\}^\{n-1\}\; lk\_l\; =\; n\; -\; 1$

The formula can be rewritten in terms of complete Bell polynomials of arguments $t\_l\; =\; -\; (l\; -\; 1)!\; \backslash operatorname\{tr\}\backslash left(A^l\backslash right)$ as

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash frac\{1\}\{\backslash det(\backslash mathbf\{A\})\}\; \backslash sum\_\{s=1\}^n\; \backslash mathbf\{A\}^\{s-1\}\; \backslash frac\{(-1)^\{n\; -\; 1\}\}\{(n\; -\; s)!\}\; B\_\{n-s\}(t\_1,\; t\_2,\; \backslash ldots,\; t\_\{n-s\})$

That is described in more detail under Cayley–Hamilton method.

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

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

where is the square matrix whose th column is the eigenvector $q\_i$ of , and is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, that is, $\backslash Lambda\_\{ii\}\; =\; \backslash lambda\_i.$ If

is symmetric,  is guaranteed to be an orthogonal matrix, therefore $\backslash mathbf\{Q\}^\{-1\}\; =\; \backslash mathbf\{Q\}^\backslash mathrm\{T\}\; .$ Furthermore, because  is a diagonal matrix, its inverse is easy to calculate:
     

$\backslash left\backslash Lambda^\{-1\}\backslash right\_\{ii\}\; =\; \backslash frac\{1\}\{\backslash lambda\_i\}$

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

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash left(\backslash mathbf\{L\}^*\backslash right)^\{-1\}\; \backslash mathbf\{L\}^\{-1\}\; ,$

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

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

$\backslash mathbf\{A\}^\{-1\}\; =\; \{1\; \backslash over\; \backslash begin\{vmatrix\}\backslash mathbf\{A\}\backslash end\{vmatrix\}\}\backslash mathbf\{C\}^\backslash mathrm\{T\}\; =$

 {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

$\backslash left(\backslash mathbf\{A\}^\{-1\}\backslash right)\_\{ij\}\; =$

 {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 is the determinant of , is the matrix of cofactors, and represents the matrix transpose.

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

(2025). 9780961408893, SIAM. . ISBN 9780961408893

, Chapter 2, page 71

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

   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 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

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash frac\{1\}\{\backslash det\; \backslash mathbf\{A\}\}\; \backslash left$

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Inversion of 3 × 3 matrices A computationally efficient matrix inversion is given by

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

   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 is not to be confused with the matrix ).

If the determinant is non-zero, the matrix is invertible, with the entries of the intermediary matrix on the right side above given by

$\backslash begin\{alignat\}\{6\}$

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{alignat}

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

$\backslash det(\backslash mathbf\{A\})\; =\; aA\; +\; bB\; +\; cC$

The Cayley–Hamilton decomposition gives

$\backslash mathbf\{A\}^\{-1\}\; =$

 \frac{1}{\det (\mathbf{A})}\left( \tfrac{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 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix (consisting of three column vectors, $\backslash mathbf\{x\}\_0$, $\backslash mathbf\{x\}\_1$, and $\backslash mathbf\{x\}\_2$) is invertible, its inverse is given by

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash frac\{1\}\{\backslash det(\backslash mathbf\; A)\}\backslash begin\{bmatrix\}$

 {(\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 , , is equal to the triple product of , , and —the volume of the parallelepiped formed by the rows or columns:

$\backslash det(\backslash mathbf\{A\})\; =\; \backslash mathbf\{x\}\_0\backslash cdot(\backslash mathbf\{x\}\_1\backslash times\backslash mathbf\{x\}\_2)$

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 is orthogonal to the non-corresponding two columns of (causing the off-diagonal terms of $\backslash mathbf\{I\}\; =\; \backslash mathbf\{A\}^\{-1\}\backslash mathbf\{A\}$ be zero). Dividing by

$\backslash det(\backslash mathbf\{A\})\; =\; \backslash mathbf\{x\}\_0\backslash cdot(\backslash mathbf\{x\}\_1\backslash times\backslash mathbf\{x\}\_2)$

causes the diagonal entries of to be unity. For example, the first diagonal is:

$1\; =\; \backslash frac\{1\}\{\backslash mathbf\{x\_0\}\backslash cdot(\backslash mathbf\{x\}\_1\backslash times\backslash mathbf\{x\}\_2)\}\; \backslash mathbf\{x\_0\}\backslash cdot(\backslash mathbf\{x\}\_1\backslash times\backslash mathbf\{x\}\_2)$

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

$\backslash begin\{align\}$

\mathbf{A}^{-1} =

 \frac{1}{\det(\mathbf{A})}\Bigl(
   &\tfrac{1}{6}\bigl( (\operatorname{tr}\mathbf{A})^{3} - 3\operatorname{tr}\mathbf{A}\operatorname{tr}(\mathbf{A}^{2}) + 2\operatorname{tr}(\mathbf{A}^{3})\bigr) \mathbf{I} \\[-3mu]
     

&\ \ \ - \tfrac{1}{2}\mathbf{A}\bigl((\operatorname{tr}\mathbf{A})^{2} - \operatorname{tr}(\mathbf{A}^{2})\bigr) + \mathbf{A}^{2}\operatorname{tr}\mathbf{A} -

   \mathbf{A}^{3}
 \Bigr)
     

\end{align}

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Blockwise inversion Let

where , , and are block matrix of arbitrary size and $\backslash mathbf\; M\; /\; \backslash mathbf\; A\; :=\; \backslash mathbf\; D\; -\; \backslash mathbf\; C\; \backslash mathbf\; A^\{-1\}\; \backslash mathbf\; B$ is the Schur complement of . ( must be square, so that it can be inverted. Furthermore, and must be nonsingular.

)

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

The strategy is particularly advantageous if is diagonal and is a small matrix, since they are the only matrices requiring inversion.

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

The inversion procedure that led to Equation () performed matrix block operations that operated on and first. Instead, if and are operated on first, and provided and are nonsingular,

the result is

Equating the upper-left sub-matrices of Equations () and () leads to

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

If and are both invertible, then the above two block matrix inverses can be combined to provide the simple factorization

By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

This formula simplifies significantly when the upper right block matrix is the zero matrix. This formulation is useful when the matrices and 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

 \begin{bmatrix} \mathbf{A}^{-1} & \mathbf{0} \\ -\mathbf{D}^{-1}\mathbf{CA}^{-1} & \mathbf{D}^{-1} \end{bmatrix}
     

If the given invertible matrix is a symmetric matrix with invertible block the following block inverse formula holds

where $\backslash mathbf\{S\}\; =\; \backslash mathbf\{D\}\; -\; \backslash mathbf\{C\}\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{C\}^T$. This requires 2 inversions of the half-sized matrices and and only 4 multiplications of half-sized matrices, if organized properly together with some additions, subtractions, negations and transpositions of negligible complexity. Any matrix $\backslash mathbf\{M\}$ has an associated positive semidefinite, symmetric matrix $\backslash mathbf\{M\}^T\backslash mathbf\{M\}$, which is exactly invertible (and positive definite), if and only if $\backslash mathbf\{M\}$ is invertible. By writing $\backslash mathbf\{M\}^\{-1\}=\backslash left(\backslash mathbf\{M\}^T\backslash mathbf\{M\}\backslash right)^\{-1\}\backslash mathbf\{M\}^T$ matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the positive definite matrix $\backslash mathbf\{M\}^T\backslash mathbf\{M\}$ satisfies the invertibility condition for its left upper block .

Those formulas together allow to construct a divide and conquer algorithm that uses blockwise inversion of associated symmetric matrices to invert a matrix with the same time complexity as the matrix multiplication algorithm that is used internally.T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, MA, 2009, §28.2. Research into matrix multiplication complexity shows that there exist matrix multiplication algorithms with a complexity of operations, while the best proven lower bound is .Ran Raz. On the complexity of matrix product. In Proceedings of the thirty-fourth annual ACM symposium on Theory of computing. ACM Press, 2002. .

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By Neumann series If a matrix has the property that

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

then is nonsingular and its inverse may be expressed by a Neumann series:

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

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

$\backslash sum\_\{n=0\}^\{2^L-1\}\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^n\; =\; \backslash prod\_\{l=0\}^\{L-1\}\backslash left(\backslash mathbf\; I\; +\; (\backslash mathbf\; I\; -\; \backslash mathbf\; A)^\{2^l\}\backslash right)$

Therefore, only matrix multiplications are needed to compute terms of the sum.

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

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

then is nonsingular and its inverse is

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

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

$\backslash mathbf\; A^\{-1\}\; =\; \backslash mathbf\; X^\{-1\}\; -\; \backslash frac\{\backslash mathbf\; X^\{-1\}\; (\backslash mathbf\; A\; -\; \backslash mathbf\; X)\; \backslash mathbf\; X^\{-1\}\}\{1\; +\; \backslash operatorname\{tr\}\backslash left(\backslash mathbf\; X^\{-1\}\; (\backslash mathbf\; A\; -\; \backslash mathbf\; X)\backslash right)\}~$

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p-adic approximation If is a matrix with integer or rational number entries, and we seek a solution in arbitrary-precision rationals, a p-adic approximation method converges to an exact solution in , assuming standard matrix multiplication is used. The method relies on solving linear systems via Dixon's method of -adic approximation (each in ) and is available as such in software specialized in arbitrary-precision matrix operations, for example, in IML.

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Reciprocal basis vectors method Given an square matrix $\backslash mathbf\{X\}\; =\; \backslash left$, $1\; \backslash leq\; i,j\; \backslash leq\; n$, with rows interpreted as vectors $\backslash mathbf\{x\}\_\{i\}\; =\; x^\{ij\}\; \backslash mathbf\{e\}\_\{j\}$ (Einstein summation assumed) where the $\backslash mathbf\{e\}\_\{j\}$ are a standard orthonormal basis of Euclidean space $\backslash mathbb\{R\}^\{n\}$ ($\backslash mathbf\{e\}\_\{i\}\; =\; \backslash mathbf\{e\}^\{i\},\; \backslash mathbf\{e\}\_\{i\}\; \backslash cdot\; \backslash mathbf\{e\}^\{j\}\; =\; \backslash delta\_i^j$), then using Clifford algebra (or geometric algebra) we compute the reciprocal (sometimes called dual) column vectors:

$\backslash mathbf\{x\}^\{i\}\; =\; x\_\{ji\}\; \backslash mathbf\{e\}^\{j\}\; =\; (-1)^\{i-1\}\; (\backslash mathbf\{x\}\_\{1\}\; \backslash wedge\backslash cdots\backslash wedge\; ()\_\{i\}\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\{x\}\_\{n\})\; \backslash cdot\; (\backslash mathbf\{x\}\_\{1\}\; \backslash wedge\backslash \; \backslash mathbf\{x\}\_\{2\}\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\{x\}\_\{n\})^\{-1\}$

as the columns of the inverse matrix $\backslash mathbf\{X\}^\{-1\}\; =\; x\_\{ji\}.$ Note that, the place "$()\_\{i\}$" indicates that "$\backslash mathbf\{x\}\_\{i\}$" is removed from that place in the above expression for $\backslash mathbf\{x\}^\{i\}$. We then have $\backslash mathbf\{X\}\backslash mathbf\{X\}^\{-1\}\; =\; \backslash left\; =\; \backslash left\; =\; \backslash mathbf\{I\}\_\{n\}$, where $\backslash delta\_\{i\}^\{j\}$ is the Kronecker delta. We also have $\backslash mathbf\{X\}^\{-1\}\backslash mathbf\{X\}\; =\; \backslash left\backslash left(\backslash mathbf\{e\}\_\{i\}\backslash cdot\backslash mathbf\{x\}^\{k\}\backslash right)\backslash left(\backslash mathbf\{e\}^\{j\}\backslash cdot\backslash mathbf\{x\}\_\{k\}\backslash right)\backslash right\; =\; \backslash left\backslash mathbf\{e\}\_\{i\}\backslash cdot\backslash mathbf\{e\}^\{j\}\backslash right\; =\; \backslash left\backslash delta\_\{i\}^\{j\}\backslash right\; =\; \backslash mathbf\{I\}\_\{n\}$, as required. If the vectors $\backslash mathbf\{x\}\_\{i\}$ are not linearly independent, then $(\backslash mathbf\{x\}\_\{1\}\; \backslash wedge\; \backslash mathbf\{x\}\_\{2\}\; \backslash wedge\backslash cdots\backslash wedge\backslash mathbf\{x\}\_\{n\})\; =\; 0$ and the matrix $\backslash mathbf\{X\}$ is not invertible (has no inverse).

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Properties

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Singularity Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero.

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Invertible matrix theorem Let be a square -by- matrix over a field (e.g., the field of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:

• is invertible, i.e. it has an inverse under matrix multiplication, i.e., there exists a such that . (In that statement, "invertible" can equivalently be replaced with "left-invertible" or "right-invertible" in which one-sided inverses are considered.)
• The linear transformation mapping to is invertible, i.e., it has an inverse under function composition. (There, again, "invertible" can equivalently be replaced with either "left-invertible" or "right-invertible".)
• The transpose is an invertible matrix.
• is Row equivalence to the -by- identity matrix .
• is Row equivalence to the -by- identity matrix .
• has .
• has full rank: .
• has a trivial kernel:
• The linear transformation mapping to is bijective; that is, the equation has exactly one solution for each in . (There, "bijective" can equivalently be replaced with "injective" or "surjective".)
• The columns of form a basis of . (In this statement, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
• The rows of form a basis of . (Similarly, here, "basis" can equivalently be replaced with either "linearly independent set" or "spanning set")
• The determinant of is nonzero: . In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit (i.e. multiplicatively invertible element) of that ring.
• The number 0 is not an eigenvalue of . (More generally, a number $\backslash lambda$ is an eigenvalue of if the matrix $\backslash mathbf\{A\}-\backslash lambda\; \backslash mathbf\{I\}$ is singular, where is the identity matrix.)
• The matrix can be expressed as a finite product of elementary matrices.

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

• $(\backslash mathbf\; A^\{-1\})^\{-1\}\; =\; \backslash mathbf\; A$
• $(k\; \backslash mathbf\; A)^\{-1\}\; =\; k^\{-1\}\; \backslash mathbf\; A^\{-1\}$ for nonzero scalar
• $(\backslash mathbf\{Ax\})^+\; =\; \backslash mathbf\; x^+\; \backslash mathbf\; A^\{-1\}$ if has orthonormal columns, where denotes the Moore–Penrose inverse and is a vector
• $(\backslash mathbf\; A^\backslash mathrm\{T\})^\{-1\}\; =\; (\backslash mathbf\; A^\{-1\})^\backslash mathrm\{T\}$
• For any invertible -by- matrices and , $(\backslash mathbf\{AB\})^\{-1\}\; =\; \backslash mathbf\; B^\{-1\}\; \backslash mathbf\; A^\{-1\}.$ More generally, if $\backslash mathbf\; A\_1,\; \backslash dots,\; \backslash mathbf\; A\_k$ are invertible -by- matrices, then $(\backslash mathbf\; A\_1\; \backslash mathbf\; A\_2\; \backslash cdots\; \backslash mathbf\; A\_\{k-1\}\; \backslash mathbf\; A\_k)^\{-1\}\; =\; \backslash mathbf\; A\_k^\{-1\}\; \backslash mathbf\; A\_\{k-1\}^\{-1\}\; \backslash cdots\; \backslash mathbf\; A\_2^\{-1\}\; \backslash mathbf\; A\_1^\{-1\}.$
• $\backslash det\; \backslash mathbf\; A^\{-1\}\; =\; (\backslash det\; \backslash mathbf\; A)^\{-1\}.$
• Left and right inverses are equal. That is, if $\backslash mathbf\{LA\}\; =\; \backslash mathbf\; I$ and $\backslash mathbf\{AR\}\; =\; \backslash mathbf\; I$ then $\backslash mathbf\; L\; =\; \backslash mathbf\; L(\backslash mathbf\{AR\})\; =\; (\backslash mathbf\{LA\})\; \backslash mathbf\; R\; =\; \backslash mathbf\; R$.

The rows of the inverse matrix of a matrix are orthonormal to the columns of (and vice versa interchanging rows for columns). To see this, suppose that where the rows of are denoted as $v\_i^\{\backslash mathrm\{T\}\}$ and the columns of as $u\_j$ for $1\; \backslash leq\; i,j\; \backslash leq\; n.$ Then clearly, the Dot product of any two $v\_i^\{\backslash mathrm\{T\}\}\; u\_j\; =\; \backslash delta\_\{i,j\}.$ 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 are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse .

A matrix that is its own inverse (i.e., a matrix such that and consequently ) is called an involutory matrix.

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In relation to its adjugate The Adjugate matrix of a matrix can be used to find the inverse of as follows:

If is an invertible matrix, then

$\backslash mathbf\{A\}^\{-1\}\; =\; \backslash frac\{1\}\{\backslash det(\backslash mathbf\{A\})\}\; \backslash operatorname\{adj\}(\backslash mathbf\{A\})$

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In relation to the identity matrix It follows from the associativity of matrix multiplication that if

$\backslash mathbf\{AB\}\; =\; \backslash mathbf\{I\}\; \backslash$

for finite square matrices and , then also

$\backslash mathbf\{BA\}\; =\; \backslash mathbf\{I\}\backslash$

(1985). 9780521386326, Cambridge University Press. ISBN 9780521386326

.

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

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

In practice, however, non-invertible matrices may be encountered. In numerical calculations, matrices that are invertible but close to a non-invertible matrix may still be problematic and are said to be ill-conditioned.

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

(1999). 047198633X, John Wiley & Sons. 047198633X

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

To derive the above expression for the derivative of the inverse of A, one can differentiate the definition of the matrix inverse $\backslash mathbf\{A\}^\{-1\}\backslash mathbf\{A\}=\backslash mathbf\{I\}$ using the product rule, and then solve for the derivative of the inverse of A:

$$

   \mathbf{0}
 = \frac{\mathrm{d}\mathbf{I}}{\mathrm{d}t}
 = \frac{\mathrm{d}(\mathbf{A}^{-1}\mathbf{A})}{\mathrm{d}t}
 = \frac{\mathrm{d}(\mathbf{A}^{-1})}{\mathrm{d}t}\mathbf{A}
   + \mathbf{A}^{-1}\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}
     

Subtracting $\backslash mathbf\{A\}^\{-1\}\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{A\}\}\{\backslash mathrm\{d\}t\}$ from both ends of this formula, and multiplying on the right by $\backslash mathbf\{A\}^\{-1\}$ finishes the derivation.

If $\backslash varepsilon$ is a small number then the derivative formula gives:

$\backslash left(\backslash mathbf\{A\}\; +\; \backslash varepsilon\backslash mathbf\{X\}\backslash right)^\{-1\}$

 = \mathbf{A}^{-1} - \varepsilon \mathbf{A}^{-1} \mathbf{X} \mathbf{A}^{-1} + \mathcal{O}(\varepsilon^2)\,
     

Given a positive integer $n$,

$$

\begin{align} \frac{ \mathrm{d}}{ \mathrm{d}t} \mathbf{A}^{n} &=

  \sum_{i=1}^n \mathbf{A}^{i-1}\frac{ \mathrm{d}\mathbf{A}}{ \mathrm{d}t}\mathbf{A}^{n-i},\\
     

\frac{ \mathrm{d}}{ \mathrm{d}t} \mathbf{A}^{-n} &=

 -\sum_{i=1}^n \mathbf{A}^{-i}\frac{ \mathrm{d}\mathbf{A}}{ \mathrm{d}t}\mathbf{A}^{-(n+1-i)}
     

\end{align}

In particular,

$$

\begin{align} (\mathbf{A} + \varepsilon \mathbf{X})^{n} &=

 \mathbf{A}^{n} + \varepsilon \sum_{i=1}^n \mathbf{A}^{i-1}\mathbf{X}\mathbf{A}^{n-i} + \mathcal{O}\left(\varepsilon^2\right),\\
     

(\mathbf{A} + \varepsilon \mathbf{X})^{-n} &=

 \mathbf{A}^{-n} - \varepsilon \sum_{i=1}^n \mathbf{A}^{-i}\mathbf{X}\mathbf{A}^{-(n+1-i)} + \mathcal{O}\left(\varepsilon^2\right)
     

\end{align}

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Generalizations

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Non-square matrices Non-square matrices, i.e. -by- matrices for which , do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is -by- and the rank of is equal to , (), then has a left inverse, an -by- matrix such that . If has rank (), then it has a right inverse, an -by- matrix such that .

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

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In Abstract algebra While the most common case is that of matrices over the real number or complex number numbers, all of those definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a ring being Commutative ring, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.

The set of invertible matrices together with the operation of matrix multiplication and entries from ring form a group, the general linear group of degree , denoted .

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Applications For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary for the matrix involved to be 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.

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Regression/least squares Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy and is 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.

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Matrix inverses in real-time simulations 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.

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Matrix inverses in MIMO wireless communication 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 so that the receiver can figure out the transmitted information.

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See also

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Further reading

• Dennis S. Bernstein (2025). 9780691140391, Princeton University Press. . ISBN 9780691140391

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External links

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