Invertible Matrix - Properties

Properties

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:

A is invertible.

A is row-equivalent to the n-by-n identity matrix I_n.

A is column-equivalent to the n-by-n identity matrix I_n.

A has n pivot positions.

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.

rank A = n.

The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0})

The equation Ax = b has exactly one solution for each b in Kn.

The columns of A are linearly independent.

The columns of A span Kn (i.e. Col A = Kn).

The columns of A form a basis of Kn.

The linear transformation mapping x to Ax is a bijection from Kn to Kn.

There is an n by n matrix B such that AB = I_n = BA.

The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn).

The number 0 is not an eigenvalue of A.

The matrix A can be expressed as a finite product of elementary matrices.

Furthermore, the following properties hold for an invertible matrix A:

• (A−1)−1 = A;
• (kA)−1 = k−1A−1 for nonzero scalar k;
• (AT)−1 = (A−1)T;
• For any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1. More generally, if A₁,...,A_k are invertible n-by-n matrices, then (A₁A₂⋯A_k−1A_k)−1 = A_k−1A_k−1−1⋯A₂−1A₁−1;
• det(A−1) = det(A)−1.

A matrix that is its own inverse, i.e. A = A-1 and A2 = I, is called an involution.