# Composition of Matrix relation I am unsure if this exercise is possible to do, could anyone tell me

Composition of Matrix relation
I am unsure if this exercise is possible to do, could anyone tell me if I am correct or not?
We have DOMAIN $\left\{1,2,3\right\}$ and RANGE $\left\{1,2,3,4\right\}$ and relation $R=\left\{\left(1,2\right),\left(2,3\right),\left(3,4\right)\right\}$.
The exercise say to find ${R}^{2}$.
I have tried two way to find this.
1. Matrix composition. Here i cam across the problem that you cannot compose a $3×4$ matrix with another $3×4$.
2. Compose the relations themselves what i get is $\left\{\left(1,3\right),\left(2,4\right),\left(3,?\right)\right\}$
I cant see how i can find the 4 to replace ?
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Step 1
Interpretation 2 is correct (interpretation 1 is not). According to the prescription for R,
.
Then, if we compose R with itself, we get
.
Step 2
That is, in the last spot, since 4 is not in the domain of R, we can't plug 3 into ${R}^{2}$. Thus, 3 is not in the domain of ${R}^{2}$, and the relation ${R}^{2}$ is defined by the prescription

In the language of ordered pairs, ${R}^{2}=\left\{\left(1,3\right),\left(2,4\right)\right\}\phantom{\rule{thinmathspace}{0ex}}.$.
The domain is the set $\left\{1,2\right\}$, and the range is the set $\left\{3,4\right\}$.

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