There is a practice problem for Chapter 3 of Prof. Strang's famous open linear algebra course (problem 7 from problem set 3). It seems the answer is incorrect, and it reveal's something I have a question about.
$$R=\left [
\begin{array}{}
I & F \\
0 & 0
\end{array}
\right ]$$
$R$ is the reduced row echelon form (rref) of some matrix $A$. $F$ represents the columns of free variables and could take any values. The question is to show that the rref of $R^TR$ is $R$. Here is the solution, I think the frist part is incorrect.
That doesn't seem right to me, I think
$$R^TR=\left [
\begin{array}{}
I & F \\
F^T & F^TF
\end{array}
\right ]$$
But, just working some examples, the second part I still find correct: $rref(R^TR)=R$. So finally to my question, how can I show this last identity, going from $R^TR$ to $rref(R^TR)=R$ with matrix algebra? So I see how the identity matrix clears out the lower left corner, $F^T$, but I am not seeing how this necessarily converts the bottom right, $F^TF$, to $0$.
Best Answer
Take $$C = \begin{bmatrix} I & 0\\-F^T & I \end{bmatrix}.$$ Compute $CR^TR$. This is called elementary BLOCK row operations. It can also be written as a string of elementary row operations.