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.

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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 $0$.


1 Answer 1


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.

  • $\begingroup$ I would think the purpose of this is to get $CR^T=I$ however I am not seeing that, and i am not seeing that $CR^TR=R$. $\endgroup$ Jan 10, 2020 at 3:38
  • $\begingroup$ Have you actually done the operation $C(R^TR)$? $\endgroup$ Jan 10, 2020 at 3:41
  • $\begingroup$ Sorry, my bad. I changed $C$. $\endgroup$ Jan 10, 2020 at 3:46
  • $\begingroup$ OK yes now I see it, I should have known. It is actually just 1 step of elimination just like the case of a 2x2 matrix. Thanks for the help! $\endgroup$ Jan 10, 2020 at 3:50

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