Setting corresponding entries in a matrix I've recently read "Matrix Inversion and the Great Injustice", a rather humorous article of a student venting his frustrations due to feeling as if he has been graded unfairly.
I follow everything so far, up until this part (at the bottom of the first page): 
I was asked to solve the following problem:
Determine the inverse matrix of:  \begin{bmatrix}1 & 3\\1 & 1\end{bmatrix} 
I remembered that the inverse matrix must be a matrix.
$$ X = \begin{bmatrix}\mathbf{x}_1 & \mathbf{x}_2\\\mathbf{x}_3 & \mathbf{x}_4\end{bmatrix} $$
of size 2 × 2 such that
$$MX = I$$
where I is the identity matrix. When I multiplied the left hand side I got
$$ \begin{bmatrix}\mathbf{x}_1 + 3\mathbf{x}_3 & \mathbf{x}_2 + 3\mathbf{x}_4\\\mathbf{x}_1  + \mathbf{x}_3 & \mathbf{x}_2 + \mathbf{x}_4\end{bmatrix} = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} $$
and when I set the corresponding entries equal I obtained the following system
\begin{bmatrix}1 & 0 & 3 & 0  & | & 1\\0 & 1 & 0 & 3  &| & 0 \\1 & 0 &1 & 0 & | & 0\\ 0 & 1 & 0 & 1 & | & 1\end{bmatrix}
Now, I do not understand what is meant with "setting the corresponding entries equal".  Can someone please point me in the right direction?
Appreciate it.

It means that if you have two matrices:
$\begin{bmatrix}
a&b\\c&d
\end{bmatrix}=\begin{bmatrix}
h&i\\j&k
\end{bmatrix}$
then it must be the case that $a=h, b=i, c=j, d=k$.
In your case,
$$\begin{bmatrix}\mathbf{x}_1 + 3\mathbf{x}_3 & \mathbf{x}_2 + 3\mathbf{x}_4\\\mathbf{x}_1  + \mathbf{x}_3 & \mathbf{x}_2 + \mathbf{x}_4\end{bmatrix} = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$
so $a=h$, where $a=x_1+3x_3$ and $h=1$. You can write this as an equation: $x_1+0x_2+3x_3+0x_4=1$. The process is similar for the other entries.
