Other uses for elementary matrices

To check that a matrix B is the inverse of a matrix A, the definition says you have to check two conditions: AB = I and BA = I. It turns out that you need check only one of these conditions; the other follows automatically. Here's why.

If A and B are square matrices with AB = I, then BA = I.

Proof: Suppose A has reduced row-echelon form R; then for some elementary matrices E1, E2, ..., Ek, we have that Ek...E2E1A = R. Set C = Ek...E2E1; then C is invertible and CA = R.

Multiply on the right by BC–1 and simplify:
          CABC–1 = RBC–1
            CIC–1 = RBC–1,
                   I = R(BC–1).

Since I doesn't have a row of zeroes, neither can R, so R must be the identity matrix.

Since R was the reduced row-echelon form of A, A is invertible.

Since AB = I, A–1(AB)A = A–1IA, i.e. BA = I.

In some applications of matrices, it's useful to be able to decompose a matrix into the product of simpler matrices.

The square matrix A can be written as the product of elementary matrices if and only if A is invertible.

Proof: If A is the product of elementary matrices, then it is invertible, since each elementary matrix is invertible.

If A is invertible, then Ek...E2E1A = I for some elementary matrices E1, E2, ..., Ek. Solve for A: (Ek...E2E1)–1(Ek...E2E1)A = (Ek...E2E1)–1I, so A = (Ek...E2E1)–1 = E1–1E2–1...Ek–1. A is then the product of the elementary matrices E1–1, E2–1, ..., Ek–1.

This gives you a method for decomposing any invertible matrix into elementary matrices.

To decompose an invertible matrix A into the product of elementary matrices:

Row reduce A to I, and record the row operations in the order you use them.

Write down the inverse row operations in the same order.

Write down the product of the elementary matrices corresponding to these inverse row operations in the same order.