I just bought the first edition of Artin's Algebra, and I'm already finding the notation being used for matrices in the first chapter extremely confusing.
On page $6$, he states the following
Here are some shorthand ways of drawing the matrix $I_n$. $$ I_n=\begin{bmatrix} 1 && && 0 \\ && \ddots && \\ 0 && && 1 \end{bmatrix} = \begin{bmatrix} 1 && && \\ && \ddots && \\ && && 1 \end{bmatrix} $$ We often indicate that a whole region in a matrix consists of zeros by leaving it blank or by putting in a single 0.
To me, this seems fairly straight forward, and the notation seems to have adequate motivation. Later though, he starts leaving blank spaces in matrices where it seems unnecessary. For example, on page $7$, he writes $$ \begin{bmatrix} 1 && \\ && 2 \end{bmatrix}^{-1} = \begin{bmatrix} 1 && \\ && \frac{1}{2} \end{bmatrix} $$
instead of the more readable $$ \begin{bmatrix} 1 && 0 \\ 0 && 2 \end{bmatrix}^{-1} = \begin{bmatrix} 1 && 0 \\ 0 && \frac{1}{2} \end{bmatrix}. $$ I don't see any motivation to using this notation, as it is clearly not difficult to put two extra numbers in each matrix. Moreover, this notation is inconsistent, as sometimes he chooses not to omit the zeros. For example, on page $10$, he writes $$ \begin{bmatrix} 0 && -1 && 2 \\ 3 && 4 && -6 \end{bmatrix} \begin{bmatrix} 1 && 0 \\ 4 && 2 \\ 3 && 2 \end{bmatrix} = \begin{bmatrix} 2 && 2 \\ 1 && -4 \end{bmatrix} $$
My questions are: What is the motivation behind leaving blanks to represent zeros on small, easily displayed matrices? Is there a reason this notation is inconsistent throughout the book, or am I simply misinterpreting the notation? Lastly, is this notation commonly used, or is this just Artin's preference?
I want to make sure when I start working through this book, notational inconvenience doesn't prevent my understanding, so any help is much appreciated.