Transforms and Inverses

As stated on the previous page, AT-1 = A-1T. And a demonstration is in order.

The new equations

The original equations will not suffice, because we need an invertible equation set. For this demonstration we'll use these:

In matrix notation this is:

Where the matrix A is:

Inverting this new matrix

Augment the matrix with identity and begin the row operations:

Subtract row 1 from row 3.

Divide through row 3 by -2.

Add row 3 to row 2.

Subtract row 3 from row 1.

The inverse is:

Proving the inverse is correct

This inverse is apparently the correct one. Now let's generate this solution from the transpose of the original matrix.

Inverting the transpose

The transpose of the matrix A at the top of this page is (AT):

Augment and begin the row operations.

Subtract row 1 from row 3.

Add row 2 to row 3.

Divide through row 3 by -2.

Subtract row 3 from row 1.

The inverse of AT (designated AT-1) is:

The transpose of this matrix (AT-1T) is:

And this is the result calculated above for the stated problem. We demonstrated the transpose of the inverse of the transpose is the inverse of the matrix.

In the next section we'll use column operations instead of doing the row operations on the transposed matrix.