In linear algebra, an -by- square matrix is called invertible (also nonsingular, nondegenerate or —rarely used— regular), if there exists an -by- square matrix such thatwhere denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) inverse of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix .
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. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded 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. 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 .
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 algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a 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 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 .
Properties
The 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: The matrix has a left inverse under matrix multiplication (that is, there exists a such that ) The matrix has a right inverse under matrix multiplication (that is, there exists a such that ). is invertible, i.e. it has an inverse under matrix multiplication (that is, there exists a such that ). The linear transformation mapping to has a left inverse under function composition. The linear transformation mapping to has a right inverse under function composition. The linear transformation mapping to is invertible, that is, has an inverse under function composition. is row-equivalent to the -by- identity matrix . is column-equivalent to the -by- identity matrix . has pivot positions. has full rank: . has a trivial kernel: The linear transformation mapping to is surjective; that is, the equation has at least one solution for each in . The linear transformation mapping to is injective; that is, the equation has at most one solution for each in . The linear transformation mapping to is bijective; that is, the equation has exactly one solution for each in . The columns of are linearly independent. The rows of are linearly independent. The columns of span . The rows of span . The columns of form a basis of . The rows of form a basis of . The determinant of is nonzero: . (In general, a square matrix over a commutative ring is invertible if and only if its determinant is an element of that ring with a multiplicative inverse (i.e. a unit) The number 0 is not an eigenvalue of . (More generally, a number is an eigenvalue of if the matrix is singular, where is the identity matrix.) The transpose is an invertible matrix. The matrix can be expressed as a finite product of elementary matrices.
Other properties
Furthermore, the following properties hold for an invertible matrix :
for nonzero scalar
if has orthonormal columns, where denotes the Moore–Penrose inverse and is a vector
For any invertible -by- matrices and , More generally, if are invertible -by- matrices, then