What does a matrix being an element of another matrix supposed to mean?

Question

Lets say we have a matrix A = $$ \begin{matrix} -1 & 1\\ 2 & 1\\ \end{matrix} $$

and C = $$ \begin{matrix} 1 & 2\\ -1 & 1\\ \end{matrix} $$

The questions are:

(a) Find the invertible matrix X such that C = XA and express X as a product of two elementary matrices.
(b) Using the results in (a), find the invertible matrix N ϵ X_2x2 such that A = NC.

So what I know so far is that M would just be the inverse of A multiplied by the C matrix. And I found X to be X= $$ \begin{matrix} -2/3 & -1/3\\ 1/3 & 5/3\\ \end{matrix} $$

But after this, im just lost.

How would I express M as a product of two elementary matrices if reducing X to its RREF would require around 3 elementary row operation?
What does N ∈ M(2x2) supposed to mean?

What i'm thinking of trying to do is to augment a 2x2 identity matrix to the X matrix then perform 1 elementary operation. That will be my two elementary matrix. To illustrate:

$$[\begin{matrix}1 & 0\\0 & 1\\\end{matrix}] *E1 = [\begin{matrix} -2/3 & -1/3\\ 1/3 & 5/3\\ \end{matrix}]$$

But then if i do one elementary operation then multiply the resulting 2x2 identity matrix and the modified X, i don't get X back.

Is this how its supposed to work? The problem specified that I only use two elementary matrices but then if the RREF on the matrix, it would take more than 1 step so i'm not sure how to do this.

Please do not add links to outside pages, people here would rather not click unknown links. Either write the problem here in mathjax or put it as an image (the first would be better). — GSofer, Oct 03 '20 at 05:30
$N \in M(2 \times 2)$ means $N$ is a member of the set of all matrices of dimensions $2 \times 2$. So $N$ is a two by two matrix. It's just a "fancy" way of saying that in symbols. — Henno Brandsma, Oct 03 '20 at 07:00
@HennoBrandsma Why did the problem say to use the results that we obtained in letter a? — , Oct 03 '20 at 07:36
We "solved" $C=XA$, and for b) you need $A=XC$ so we left multiply $C$ by the two inverse elementary matrices from a) in reverse order. Elementary matrices are easily inverted. I think that's the idea. — Henno Brandsma, Oct 03 '20 at 07:44
@HennoBrandsma Ohh i see! I get it now, thanks. But now my problem is getting the two elementary matrices to represent X. What's wrong with the process that I did I specified in my post? Thank you in advance! — , Oct 03 '20 at 07:57
You need several elementary row operations to get from a matrix to its inverse, not just one.. — Henno Brandsma, Oct 03 '20 at 08:06
@HennoBrandsma But i dont understand why the question is just asking for TWO elementary matrices? — , Oct 03 '20 at 08:07

Henno Brandsma · Accepted Answer · 2020-10-03T08:48:22.673

1

Hint: See how you can change $A$ into $C$ with swaps (and negating a row)? (these correspond to elementary matrices...)

edited Oct 03 '20 at 08:48

answered Oct 03 '20 at 08:15

Henno Brandsma

242,131

Oh yeaaah, how did i miss that... Thank you so much! – Oct 03 '20 at 08:22
Ah wait. Additional question. How does doing that be equal to X? – Oct 03 '20 at 08:25
@Duts Every transform is a left multiplication with an elementary matrix. So if $X_1, X_2$ are these operations we have $C= X_2 (X_1 A)$ so we can take $X=X_2X_1$. – Henno Brandsma Oct 03 '20 at 08:32
Now i'm just lost, sorry. Which of the matrices do I exactly transform? And to what purpose? Could you tell me more about the X2 and X1 transform? – Oct 03 '20 at 08:39
Oh wait, nvm for the additional question. I get it now! Thank you very much! – Oct 03 '20 at 08:45
I wonder if the identity matrix counts as an elementary matrix. – DanielV Oct 03 '20 at 08:50
@DanielV Yes, but you don't use it normally. – Henno Brandsma Oct 03 '20 at 08:51
@HennoBrandsma Hi, could I ask your assistance here, please? Excuse me for the bother. – Antonio Maria Di Mauro Oct 03 '20 at 08:52
@HennoBrandsma Hmm with your edit, i also noticed that it requires another step which is to negate a row. Then that will be a total of 3 steps? – Oct 03 '20 at 08:58
@Duts Yes, you could combine them depending on your definition of what an elementary matrix is. – Henno Brandsma Oct 03 '20 at 09:03

What does a matrix being an element of another matrix supposed to mean?

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