# Languages, students, interpretation of $AA^{\tau}\;\&\;A^{\tau}A$

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?

• 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.) Jan 11, 2020 at 20:15
• @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 $A^TA\in M_5$. Aug 29, 2020 at 8:12

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 $$j$$.