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Student $A$ speaks French & German, student $B$ speaks English, French & Italian, student $C$ speaks English, Italian & Spanish and student $D$ speaks all the languages mentioned except French. Write a matrix such that rows represent those $4$ students, while columns represent the languages they speak. If a person $i$ speaks a language $j$, set $a_{ij}$=1, otherwise set $a_{ij}=0$. Interpret the meaning of the matrices $AA^{\tau}\;\&\;A^{\tau}A$.

My attempt:

columns are: $E,F,G,I\;\&\;S$ One of my dilemmas is if the symmetry of $AA^{\tau}\;\&\;A^{\tau}A$ is relevant here and how to use what I've gotten: $$A=\begin{bmatrix}0&1&1&0&0\\1&1&0&1&0\\1&0&0&1&1\\1&0&1&1&1\end{bmatrix},A^{\tau}=\begin{bmatrix}0&1&1&1\\1&1&0&0\\1&0&0&1\\0&1&1&1\\0&0&1&1\end{bmatrix}$$ $$AA^{\tau}=\begin{bmatrix}2&1&0&1\\1&3&2&2\\0&2&3&3\\1&2&3&4\end{bmatrix},\;A^{\tau}A=\begin{bmatrix}3&1&1&3&2\\1&2&1&1&0\\1&1&2&1&1\\3&1&1&3&2\\2&0&1&2&2\end{bmatrix}$$

I've already read posts and articles on the topic of Gramian matrix, but we haven't covered it this semester yet. I translated this from Croatian word by word and thought the interpretation should be some kind of a relation or number of combinations I'm not able to see at the moment. I'm not sure if this task explicitly involves the $\text{Gramian matrix}$. How to interpret the given matrices in the context of students and languages they speak?

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  • $\begingroup$ Just adding an additional note that the "elementary transformations" part of this post is incorrect; since $A$ isn't square, the determinant of $A$ is not defined. Your conclusion doesn't follow. (You'll actually see that the determinant of $A^T A$ is zero while the determinant of $A A^T$ is non-zero. It may be good to ask yourself why that intuitively is true.) $\endgroup$
    – Roy D.
    Jan 11, 2020 at 20:15
  • $\begingroup$ @RoyD, since $\operatorname{rank}(A)=\operatorname{rank}\left(A^T\right),\quad\operatorname{rank}\left(AA^T\right),\operatorname{rank}\left(A^TA\right)\le\operatorname{rank}(A)=4$ and $AA^T\in M_4$, while $A^TA\in M_5$. $\endgroup$
    – PinkyWay
    Aug 29, 2020 at 8:12

1 Answer 1

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The $(i,i)$ entry of $AA^T$ represents the number of languages that a particular person speak.

On the other hand $(i,j)$ entry of $AA^T$ is the number of common language between student $i$ and student $j$. This is just the inner product of the $i$-th row and the $j$-th row of which only the common language would give us the product $1$ to be counted.

Similarly, $(i,i)$ entry of $A^TA$ represents the number of students that speak that language and $(i,j)$ entry would give us the number of students that speak both language $i$ and language $j$.

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