<Dd> A B = B A = I n (\ displaystyle \ mathbf (AB) = \ mathbf (BA) = \ mathbf (I) _ (n) \) </Dd> <P> where I 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 . </P> <P> 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 . </P> <P> 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 . </P>

When does the inverse of a square matrix not exist