In linear algebra, an *n*-by-*n* (square) matrix **A** is called **invertible** or **nonsingular** or **nondegenerate** if there exists an *n*-by-*n* matrix **B** such that

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

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

^{}

Non-square matrices (*m*-by-*n* matrices for which *m ≠ n*) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If **A** is *m*-by-*n* and the rank of **A** is equal to *n*, then **A** has a left inverse: an *n*-by-*m* matrix **B** such that **BA** = **I**. If **A** has rank *m*, then it has a right inverse: an *n*-by-*m* matrix **B** such that **AB** = **I**.

A square matrix that is not invertible is called **singular** or **degenerate**. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular.

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 commutative ring.
However, in this case the condition for a square matrix to be
invertible is that its determinant is invertible in the ring, which in
general is a much stricter requirement than being nonzero.

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

Icommutative ring. However, in this case the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix

A.

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** 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.
- 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}.
- There is an
*n* by *n* matrix **C** such that **CA** = **I**_{n}.
- The transpose
**A**^{T} is an invertible matrix (hence rows of **A** are linearly independent, span *K*^{n}, and form a basis of *K*^{n}).
- The number 0 is not an eigenvalue of
**A**.
- The matrix
**A** can be expressed as a finite product of elementary matrices.

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

- .
- for nonzero scalar
*k*
- For any invertible
*n*×*n* matrices **A** and **B** . More generally, if **A**_{1},...,**A**_{k} are invertible *n*×*n* matrices, then

A matrix that is its own inverse, i.e. and , is called an involution.

### Density

Over the field of real numbers, the set of singular *n*-by-*n* matrices, considered as a subset of , is a null set, i.e., has Lebesgue measure zero. This is true because singular matrices are the roots of the polynomial function in the entries of the matrix given by the determinant. Thus in the language of measure theory, almost all *n*-by-*n* matrices are invertible. Intuitively, this means that if you pick a random square matrix over the reals, the probability that it will be singular is zero; n randomly picked vectors will form a basis for a n-space with probability 1.

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.

## Methods of matrix inversion

### Gaussian 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

Main article: Cramer's rule

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:

where |**A**| is the determinant of **A**, **C**_{ij} is the matrix of cofactors, and **A**^{T} represents the matrix transpose.

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 that the matrix involved 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.

#### Inversion of 2×2 matrices

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

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.

### Blockwise inversion

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

where **A**, **B**, **C** and **D** are matrix sub-blocks of arbitrary size. (A and D must, of course, be square, so that they can be inverted. Furthermore, this is true if and only if **A** and **D**−**CA**^{−1}**B** are nonsingular ^{}). 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
inversion.

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**^{−1}**C** are nonsingular, the result is

Equating Equations (1) and (2) leads to

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

### By Neumann series

If a matrix **A** has the property that

then **A** is nonsingular and its inverse may be expressed by a Neumann series:^{}

Truncating the sum results in an "approximate" inverse which may be useful as a preconditioner.

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

then **A** is nonsingular and its inverse is

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

##
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

To derive the above expression for the derivative of the inverse of **A**, one can differentiate the definition of the matrix inverse and then solve for the inverse of **A**:

Subtracting from both sides of the above and multiplying on the right by gives the correct expression for the derivative of the inverse:

Similarly, if ε is a small number then