Condition for matrix inverse to exist

Definition

If A is a square matrix of order n and there exists a square matrix B of the same order such that AB = BA = Iₙ, then B is called the inverse of A and is written A⁻¹. A matrix whose determinant is non-zero is called non-singular.

A square matrix A of order n has an inverse if and only if |A| ≠ 0 (A is non-singular). We prove both directions.

Tool used: for any square matrix, A(adj A) = (adj A)A = |A| Iₙ, where adj A is the transpose of the matrix of cofactors.

Necessary condition (inverse exists ⇒ |A| ≠ 0)

Let A⁻¹ exist, so AA⁻¹ = I. Taking determinants and using |AB| = |A||B|:

|A| · |A⁻¹| = |I| = 1

The product of the two numbers is 1, so neither can be zero. Hence |A| ≠ 0.

Sufficient condition (|A| ≠ 0 ⇒ inverse exists)

Let |A| ≠ 0. From A(adj A) = (adj A)A = |A| I, divide by the non-zero scalar |A|:

A · (adj A / |A|) = (adj A / |A|) · A = I

So the matrix B = adj A / |A| satisfies AB = BA = I. By definition B is the inverse of A, so A⁻¹ = adj A / |A| exists.

Uniqueness of the inverse

If B and C are both inverses, B = BI = B(AC) = (BA)C = IC = C.

Thus A has an inverse if and only if A is non-singular. ∎