# Invertible matrix

An **invertible matrix** is a square matrix \( A \) for which there exists another matrix \( A^{-1} \) of the same size such that \( A \cdot A^{-1} = A^{-1} \cdot A = I_n \), the identity matrix; \( A^{-1} \) is called the inverse of \( A \), and a square matrix with no inverse is called singular.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup> Invertibility is the matrix analog of a nonzero number having a reciprocal, and it is equivalent to a long list of other properties of the matrix and of the linear transformation it defines.<sup>[2](https://mathworld.wolfram.com/InvertibleMatrixTheorem.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( A \) is invertible if some \( A^{-1} \) satisfies \( A \cdot A^{-1} = A^{-1} \cdot A = I_n \); the inverse is unique.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup> |
| Determinant test | Over a field, a square matrix is invertible if and only if its determinant is nonzero; a square matrix is singular only when its determinant is exactly zero.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup> |
| Equivalent conditions | Row-equivalence to the identity, \( n \) pivot positions, only the trivial solution to \( A\mathbf{x} = \mathbf{0} \), linearly independent columns, a unique solution to \( A\mathbf{x} = \mathbf{b} \) for every \( \mathbf{b} \), and a one-to-one transformation \( \mathbf{x} \mapsto A\mathbf{x} \) are all equivalent to invertibility.<sup>[2](https://mathworld.wolfram.com/InvertibleMatrixTheorem.html)</sup> |
| Ring setting | Over a commutative ring with identity, \( A \) is invertible if and only if \( \det(A) \) is invertible in the ring, with \( A^{-1} = (\det A)^{-1} \cdot \operatorname{adj} A \); a nonzero determinant is not enough.<sup>[3](https://sites.millersville.edu/bikenaga/linear-algebra/determinants-properties/determinants-properties.html)</sup> |
| Modular case | A 2 × 2 matrix over \( \mathbb{Z}/(m) \) has an inverse precisely when its determinant is invertible in \( \mathbb{Z}/(m) \); for example a matrix with determinant 8 mod 10 is not invertible even though 8 is nonzero.<sup>[4](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/modarith.pdf)</sup> |
| Numerical practice | Software libraries advise against forming the inverse explicitly: LAPACK states it is seldom necessary and not recommended for solving linear systems, and MATLAB recommends the backslash operator \( A \backslash b \) over \( \operatorname{inv}(A) \cdot b \).<sup>[5](https://www.netlib.org/lapack/lug/node26.html)</sup><sup> • </sup><sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup>  |
| History | Sylvester coined the term "matrix" in 1850; Cayley published the first inverse of a matrix by 1853 and gave an explicit construction of the inverse in terms of the determinant in his 1858 Memoir on the theory of matrices.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> |

## Definition and basic properties

The definition requires both products \( A \cdot A^{-1} \) and \( A^{-1} \cdot A \) to equal the identity. For square matrices over a field, a one-sided inverse automatically works on the other side, so it suffices to find \( B \) with \( AB = I \) or \( BA = I \).<sup>[7](https://prime.pages.ewi.tudelft.nl/openlabook-published/Appendices/InverseMatrixTheorem.html)</sup> The inverse, when it exists, is unique: if \( B \) and \( C \) are both inverses of \( A \), then \( B = B \cdot I = B(AC) = (BA)C = C \).<sup>[8](https://planetmath.org/invertiblematrix)</sup>

Invertibility behaves well under products. If \( A \) and \( B \) are invertible matrices of the same size, then \( AB \) is invertible with \( (AB)^{-1} = B^{-1} \cdot A^{-1} \), the factors reversed.

## The invertible matrix theorem

The **invertible matrix theorem** collects the many conditions equivalent to invertibility of an \( n \times n \) matrix \( A \) into a single statement. For an \( n \times n \) matrix, the following are equivalent:<sup>[2](https://mathworld.wolfram.com/InvertibleMatrixTheorem.html)</sup><sup> • </sup><sup>[7](https://prime.pages.ewi.tudelft.nl/openlabook-published/Appendices/InverseMatrixTheorem.html)</sup>

1. \( A \) is invertible.
2. \( A \) is row-equivalent to the \( n \times n \) identity matrix \( I_n \).
3. \( A \) has \( n \) pivot positions.
4. The equation \( A\mathbf{x} = \mathbf{0} \) has only the trivial solution \( \mathbf{x} = \mathbf{0} \).
5. The columns of \( A \) form a linearly independent set.
6. The columns of \( A \) span \( \mathbb{R}^n \).
7. For each \( \mathbf{b} \) in \( \mathbb{R}^n \), the equation \( A\mathbf{x} = \mathbf{b} \) has a unique solution.
8. The linear transformation \( \mathbf{x} \mapsto A\mathbf{x} \) is one-to-one (and onto).
9. \( \det(A) \neq 0 \).
10. \( 0 \) is not an eigenvalue of \( A \).
11. \( A \) has a factorization \( A = E_1 \cdots E_k \) into elementary matrices.

The theorem applies only to square matrices.<sup>[9](https://webspace.ship.edu/jehamb/ela/lecture25.html)</sup> Its practical force is that the statements stand or fall together: for invertible matrices all of them are true, and for non-invertible matrices all of them are false, so a failure of any one condition, such as a zero determinant or a nontrivial null space, rules out invertibility.<sup>[10](https://textbooks.math.gatech.edu/ila/invertible-matrix-thm.html)</sup>

## Computing the inverse

Two classical constructions give the inverse. The adjugate formula expresses

\[ A^{-1} = (\det A)^{-1} \cdot \operatorname{adj} A, \]

where \( \operatorname{adj} A \) is built from cofactors of \( A \). Cayley gave this construction in terms of the determinant in 1858.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> The formula is practical for small matrices, about 3 × 3 or smaller, but row reduction uses fewer steps for larger ones.<sup>[3](https://sites.millersville.edu/bikenaga/linear-algebra/determinants-properties/determinants-properties.html)</sup> The row-reduction method applies the operations that reduce \( A \) to \( I_n \) simultaneously to an adjoining identity matrix; the result is \( A^{-1} \).<sup>[11](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter32/section03.html)</sup>

In floating-point software, inversion is built on factorization: MATLAB's `inv` performs an [LU decomposition](https://www.edgepedia.org/lu-decomposition) of the input matrix, or an LDL decomposition if the matrix is Hermitian, and then solves linear systems to obtain the inverse.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup> SciPy's `linalg.inv` raises an error for singular input, warns on ill-conditioned inputs, and accepts structure hints such as symmetric positive definite or triangular matrices that select dedicated solvers.<sup>[12](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html)</sup>

**Why explicit inverses are avoided.** Solving \( A\mathbf{x} = \mathbf{b} \) by first forming \( A^{-1} \) and then multiplying is slower and less accurate than solving the system directly. In a MATLAB benchmark on a 500 × 500 matrix with condition number \( 10^{10} \), the backslash operator \( A \backslash b \) took about half the time of \( \operatorname{inv}(A) \cdot b \) and produced a residual error of \( 3.9285 \times 10^{-15} \) against \( 5.1611 \times 10^{-7} \) for the explicit-inverse route; the LAPACK Users' Guide states that it is seldom necessary to compute an inverse explicitly and certainly not recommended as a means of solving linear systems.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup><sup> • </sup><sup>[5](https://www.netlib.org/lapack/lug/node26.html)</sup>  Accuracy also degrades when the matrix is badly scaled or nearly singular, which is why libraries provide condition-number checks such as `rcond` and `cond`.<sup>[1](https://www.mathworks.com/help/matlab/ref/inv.html)</sup>

## Inverses over rings and modular arithmetic

Over a general commutative ring with identity, the determinant test changes form: \( A \) is invertible if and only if \( \det(A) \) is a unit of the ring, and then \( A^{-1} = (\det A)^{-1} \cdot \operatorname{adj} A \).<sup>[3](https://sites.millersville.edu/bikenaga/linear-algebra/determinants-properties/determinants-properties.html)</sup> A nonzero determinant does not suffice, because a nonzero nonunit cannot be inverted. Over \( \mathbb{Z}_6 \), for instance, the matrix with rows (5, 3) and (1, 1) has determinant 2, which is not invertible in \( \mathbb{Z}_6 \), so the matrix is not invertible despite its nonzero determinant.<sup>[3](https://sites.millersville.edu/bikenaga/linear-algebra/determinants-properties/determinants-properties.html)</sup>

The same distinction governs matrices over \( \mathbb{Z}/(m) \). A 2 × 2 matrix with entries in \( \mathbb{Z}/(m) \) has an inverse precisely when its determinant is invertible in \( \mathbb{Z}/(m) \), and the inverse is then given by the same formula as in the real case.<sup>[4](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/modarith.pdf)</sup> When \( m \) is prime, every nonzero residue is invertible, so the familiar "nonzero determinant" rule from real linear algebra carries over; for composite \( m \) it does not.<sup>[4](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/modarith.pdf)</sup>

## Applications

**Cryptography.** The Hill cipher, introduced by Lester S. Hill in 1929, encrypts blocks of letters by multiplying their number vectors by a key matrix modulo 26; the key matrix must be invertible mod 26 so that the receiver can compute its inverse and decipher.<sup>[13](https://www.cs.uri.edu/cryptography/classical-ciphers/hill/article.html)</sup> The invertibility requirement is exactly the ring condition above: the determinant of the key must be coprime to the modulus.<sup>[14](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part7/cryptography.html)</sup>

**Coordinate transformations.** In control theory and computer graphics, a change of coordinates \( \bar{\mathbf{x}} = T\mathbf{x} \) requires \( T \) to be nonsingular so that one can move back and forth between coordinate systems; the transformed system matrices become \( TAT^{-1} \), \( TB \), and \( CT^{-1} \).<sup>[15](https://courses.grainger.illinois.edu/ece486/sp2026/documentation/handbook/lec20.html)</sup> Graphics pipelines likewise use inverse matrices to undo transformations, for example taking the view matrix as the inverse of the model matrix when rendering from a moving viewpoint.<sup>[16](https://courses.grainger.illinois.edu/cs418/fa2024/text/math2.html)</sup>

## History

Determinants were studied long before matrices: Cauchy's definitive paper on determinants was read to the French Institute in 1812, and Jacobi's three 1841 treatises made the determinant widely known with an algorithmic definition.<sup>[17](https://www.nationalacademies.org/read/11540/chapter/14)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup>  Sylvester first used the term "matrix" in 1850, defining it as an oblong arrangement of terms that led to various determinants from square arrays contained within it.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> Cayley, who met Sylvester while both were practicing law in London, quickly saw the significance of the concept: by 1853 he had published a note giving, for the first time, the inverse of a matrix, and his 1858 *Memoir on the theory of matrices* contains the first abstract definition of a matrix, a matrix algebra with addition, multiplication, scalar multiplication, and inverses, and an explicit construction of the inverse in terms of the determinant.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup><sup> • </sup><sup>[17](https://www.nationalacademies.org/read/11540/chapter/14)</sup>

Frobenius's 1878 work on linear substitutions and bilinear forms proved the general Cayley-Hamilton theorem, defined the rank of a matrix, and introduced orthogonal matrices, although he did not yet use the term matrix and seemed unaware of Cayley's work.<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup> Matrix theory reached its present major role as an undergraduate topic through texts such as Bôcher's *Introduction to higher algebra* (1907) and Mirsky's *An introduction to linear algebra* (1955).<sup>[6](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)</sup>

## References

1. [inv - Matrix inverse, MATLAB documentation](https://www.mathworks.com/help/matlab/ref/inv.html)
2. [Invertible Matrix Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/InvertibleMatrixTheorem.html)
3. [Determinants - Properties, Bruce Ikenaga, Millersville University](https://sites.millersville.edu/bikenaga/linear-algebra/determinants-properties/determinants-properties.html)
4. [Modular Arithmetic, Keith Conrad, University of Connecticut](https://kconrad.math.uconn.edu/blurbs/ugradnumthy/modarith.pdf)
5. [Linear Equations, LAPACK Users' Guide](https://www.netlib.org/lapack/lug/node26.html)
6. [Matrices and determinants, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/)
7. [The inverse matrix theorem, Linear Algebra, TU Delft](https://prime.pages.ewi.tudelft.nl/openlabook-published/Appendices/InverseMatrixTheorem.html)
8. [Invertible matrix, PlanetMath](https://planetmath.org/invertiblematrix)
9. [Elementary Linear Algebra, Lecture 25: The Invertible Matrix Theorem, Shippensburg University](https://webspace.ship.edu/jehamb/ela/lecture25.html)
10. [The Invertible Matrix Theorem, Interactive Linear Algebra, Georgia Tech](https://textbooks.math.gatech.edu/ila/invertible-matrix-thm.html)
11. [The Inverse of a Matrix, MIT OCW 18.013A](https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter32/section03.html)
12. [scipy.linalg.inv, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html)
13. [Hill Cipher, University of Rhode Island cryptography course](https://www.cs.uri.edu/cryptography/classical-ciphers/hill/article.html)
14. [Hill Method, Brown University CS52 notes](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part7/cryptography.html)
15. [ECE 486 Control Systems, Lecture 20: Coordinate Transformations, University of Illinois](https://courses.grainger.illinois.edu/ece486/sp2026/documentation/handbook/lec20.html)
16. [Homogeneous Vectors and Transformations, CS 418, University of Illinois](https://courses.grainger.illinois.edu/cs418/fa2024/text/math2.html)
17. [Unknown Quantity: A Real and Imaginary History of Algebra, John Derbyshire, National Academies Press](https://www.nationalacademies.org/read/11540/chapter/14)

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