<Dl> <Dd> A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate . </Dd> <Dd> A is row - equivalent to the n - by - n identity matrix I . </Dd> <Dd> A is column - equivalent to the n - by - n identity matrix I . </Dd> <Dd> A has n pivot positions . </Dd> <Dd> 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 . </Dd> <Dd> A has full rank; that is, rank A = n . </Dd> <Dd> The equation Ax = 0 has only the trivial solution x = 0 . </Dd> <Dd> Null A = (0). </Dd> <Dd> The equation Ax = b has exactly one solution for each b in K . </Dd> <Dd> The columns of A are linearly independent . </Dd> <Dd> The columns of A span K . </Dd> <Dd> Col A = K . </Dd> <Dd> The columns of A form a basis of K . </Dd> <Dd> The linear transformation mapping x to Ax is a bijection from K to K . </Dd> <Dd> There is an n - by - n matrix B such that AB = I = BA . </Dd> <Dd> The transpose A is an invertible matrix (hence rows of A are linearly independent, span K, and form a basis of K). </Dd> <Dd> The number 0 is not an eigenvalue of A . </Dd> <Dd> The matrix A can be expressed as a finite product of elementary matrices . </Dd> <Dd> The matrix A has a left inverse (i.e. there exists a B such that BA = I) or a right inverse (i.e. there exists a C such that AC = I), in which case both left and right inverses exist and B = C = A . </Dd> </Dl> <Dd> A is invertible, i.e. A has an inverse, is nonsingular, or is nondegenerate . </Dd> <Dd> A is row - equivalent to the n - by - n identity matrix I . </Dd> <Dd> A is column - equivalent to the n - by - n identity matrix I . </Dd>

What are the conditions for a matrix to be invertible