In problem 27)e) of 3.3 we're asked to find the row reduced echelon form of (R^T)R where R is the matrix formed from [ I F ].

R^T R becomes the matrix:

| I F |
| F^T 0 |

From there however I'm unsure of how the solution reduces the top F^T to a zero submatrix. Surely if we were to eliminate using the I F rows, we would result with 0 in the bottom left sub-matrix, but the product of (F^T) F in the bottom right?

| I F |
| 0 (F^T) F |
I tried plugging in random numbers for F and confirmed this to be the case.

So how is it that the solution finds rref(R^T R) to be

I F
0 0

?

Question

In problem 27)e) of 3.3 we're asked to find the row reduced echelon form of (R^T)R where R is the matrix formed from [ I F ].

R^T R becomes the matrix:

| I      F |
| F^T 0 |

From there however I'm unsure of how the solution reduces the top F^T to a zero submatrix. Surely if we were to eliminate using the I F rows, we would result with 0 in the bottom left sub-matrix, but the product of (F^T) F in the bottom right?

| I    F          |
| 0   (F^T) F |
I tried plugging in random numbers for F and confirmed this to be the case.

So how is it that the solution finds rref(R^T R) to be

I F
0 0

?