Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that [ A B C D ] − 1 = [ A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 1 ] , {displaystyle {egin{bmatrix}mathbf {A} &mathbf {B} \mathbf {C} &mathbf {D} end{bmatrix}}^{-1}={egin{bmatrix}mathbf {A} ^{-1}+mathbf {A} ^{-1}mathbf {B} (mathbf {D} -mathbf {CA} ^{-1}mathbf {B} )^{-1}mathbf {CA} ^{-1}&-mathbf {A} ^{-1}mathbf {B} (mathbf {D} -mathbf {CA} ^{-1}mathbf {B} )^{-1}\-(mathbf {D} -mathbf {CA} ^{-1}mathbf {B} )^{-1}mathbf {CA} ^{-1}&(mathbf {D} -mathbf {CA} ^{-1}mathbf {B} )^{-1}end{bmatrix}},}     ( 1) [ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 − ( A − B D − 1 C ) − 1 B D − 1 − D − 1 C ( A − B D − 1 C ) − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 1 ] . {displaystyle {egin{bmatrix}mathbf {A} &mathbf {B} \mathbf {C} &mathbf {D} end{bmatrix}}^{-1}={egin{bmatrix}(mathbf {A} -mathbf {BD} ^{-1}mathbf {C} )^{-1}&-(mathbf {A} -mathbf {BD} ^{-1}mathbf {C} )^{-1}mathbf {BD} ^{-1}\-mathbf {D} ^{-1}mathbf {C} (mathbf {A} -mathbf {BD} ^{-1}mathbf {C} )^{-1}&quad mathbf {D} ^{-1}+mathbf {D} ^{-1}mathbf {C} (mathbf {A} -mathbf {BD} ^{-1}mathbf {C} )^{-1}mathbf {BD} ^{-1}end{bmatrix}}.}     ( 2) ( A − B D − 1 C ) − 1 = A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 {displaystyle (mathbf {A} -mathbf {BD} ^{-1}mathbf {C} )^{-1}=mathbf {A} ^{-1}+mathbf {A} ^{-1}mathbf {B} (mathbf {D} -mathbf {CA} ^{-1}mathbf {B} )^{-1}mathbf {CA} ^{-1},}     ( 3) In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In 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. 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 a square matrix randomly selected from a continuous uniform distribution on its entries will almost never be singular. 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 (n ≤ m), then A has a left inverse: an n-by-m matrix B such that BA = In. If A has rank m (m ≤ n), then it has a right inverse: an n-by-m matrix B such that AB = Im. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A. 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 left-inverse or right-inverse 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 form a group, the general linear group of degree n. Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent, that is, for any given matrix they are either all true or all false: Furthermore, the following properties hold for an invertible matrix A: