Invertible matrix
In linear algebra, an nbyn square matrix A is called invertible (also nonsingular or nondegenerate), if there exists an nbyn square matrix B such that
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where I_{n} denotes the nbyn 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 (multiplicative) inverse of A, denoted by A^{−1}.[1] Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero.[2] Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Nonsquare matrices (mbyn 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 mbyn and the rank of A is equal to n (n ≤ m), then A has a left inverse, an nbym matrix B such that BA = I_{n}. If A has rank m (m ≤ n), then it has a right inverse, an nbym matrix B such that AB = I_{m}.
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 ring. However, in the case of the ring being commutative, 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 being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of leftinverse or rightinverse are more complicated, since a notion of rank does not exist over rings.
The set of n × n invertible matrices together with the operation of matrix multiplication (and entries from ring R) form a group, the general linear group of degree n, denoted GL_{n}(R).