Singular matrix inverse with this method

We saw on the previous page that row and column operations could be applied together to invert a matrix. In this section we attempt to use this knowledge to invert the matrix from the first section (reproduced here).

A =

That matrix was singular, and the standard inverse technique failed, as did the Moore-Penrose method of multiplying the transpose of the matrix times both sides of the equation.

Augmenting for row and column operations

Augment the matrix for row and column operations:

Subtract row 1 from row 3.

Subtract column 1 from column 2.

Subtract row 2 from row 3.

Multiply through row 2 by -1.

Add column 2 to column 3.

Now that the original A matrix is reduced as much as possible, we have the task of identifying the C and R matrices and multiplying them together to get our inverse.

Creating the A-1

The full C matrix is not used. The reduced matrix is identity of rank 2, so we take the two corresponding columns of the C matrix and the two corresponding rows of the R matrix and multiply these to get our pseudo-inverse:

The reduced R matrix is:

The product of these two matrices is shown here:

Let's refer to our rather interesting inverse with the designation A‡. Having calculated it, we need a slight refresher on pseudo-inverses.

Pseudo-Inverse Refresher

I use this designation ^‡ because in the literature, a pseudo-inverse that meets all 4 of the properties uses the ^† designation. It's sort of a defacto standard. Here are the 4 properties:

1. A A† A = A
2. A† A A† = A†
3. (A A†)* = (A A†)
4. (A† A)* = (A† A)

The Hermitian properties (numbers 3 and 4 in the list) are related to pseudo-inverse being unique. The first two properties are the confirmations that the pseudo-inverse works as a left-hand or right-hand inverse.

To confirm that our inverse meets the first two criteria, we'll do the math. First we calculate (A‡ A):

Multiplying the above result by the original matrix produces the original again (property 1 above - A A‡ A = A):

And we confirm the second property (A‡ A A‡ = A‡) here:

In the next section we look at what this means to our original problem.