Undergrad resources for identifying regular languages with Myhill-Nerode matrices
I wanted to make this a comment, but it is too long.

First, your definition of the matrix is poorly written and probably wrong. There is no reason to introduce indices $i, j$. My guess is that the right definition is the following:
$T$ is the infinite matrix $(T_{u,v})_{u, v \in A^*}$ defined by
$$
  T_{u,v} = \begin{cases} 
    1 &amp;\text{if $uv \in L$}\\
    0 &amp;\text{otherwise}
            \end{cases}
$$
Next, your claim 


  $L$ is non-regular if and only if all rows of $T$ are distinct  


is wrong. Take $A = \{a, b\}$ and $L = \{ u \in A^* \mid |u|_a = |u|_b\}^1$. Then
$T_{1,v} = T_{ab,v}$ for all $v \in A^*$, so the rows $T_{1, -}$ and $T_{ab,-}$ are equal although $L$ is not regular. What is true however is that $L$ is regular if and only if $T$ has finitely many distinct rows.
${}_{\text{(1) $|u|_c$ denotes the number of occurrences of the letter $c$ in $u$.}}$

References. The matrix $T$ is used in the larger setting of noncommutative formal power series under the name of Hankel matrix. Here is a good reference, but I am afraid it is not really appropriate for undergraduate students.

[1] J. Berstel and C. Reutenauer, Noncommutative rational series with applications. Encyclopedia of Mathematics and its Applications, 137. Cambridge University Press, Cambridge, 2011. xiv+248 pp. ISBN: 978-0-521-19022-0