What is the significance of this property of the determinant?

Question

16.5. Definition.

Let $F$ be an arbitrary field. A determinant is a function which assigns to each $n$-tuple $\{a_1, \dots, a_n\}$ of vectors in $F_n$ an element of $F$, $D=D(a_1,\dots,a_n)$ such that the following conditions are satisfied.

(i) $D(a_1,\dots,a_{i-1},a_i+a_j,a_{i+1},\dots,a_n)=D(a_1,\dots,a_n)$, for $1\le i\le n$ and $j\ne i$

Why does including the $i$, $i\pm1$ and $j$ elements matter?

Answer 1

This is saying that the determinant is unchanged if you add two entries together. In terms of matrices, this means if you add one column of your matrix to another, you don't change the determinant.

The deal with the $i \pm 1$ is that they're saying $i$ is some fixed index, and everything before the $i$-th entry and everything after the $i$-th entry is unchanged -- only $i$-th entry is changed by adding the $j$-th entry.

I guess the way I usually think of determinants is that they are defined to be multilinear and alternating. This property would then be a consequence as, supposing just for notational convenience $i < j$, we have

$$D(..., a_i + a_j, ..., a_j, ...) = D(..., a_i, ..., a_j, ....) + D(..., a_j, ..., a_j, ...)$$

but this second term is zero since you have the same entry in two spots.

Answer 2

Here $a_i$ is a vector, say a row or a column of a matrix.

The determinant does not change by adding one row to another row, or by adding a column to another column.

That is what it says.

Answer 3

This is just a subtlety of using “$\ldots$” (ellipsis) notation. It’s not so much that the $i\pm1$ terms are significant, but just consider for a moment what it would look like without them:

$$D(a_1,\ldots,a_i+a_j,\ldots,a_n).$$

Notice that it’s impossible to give a reasonable interpretation for this notation. There is no sensible pattern that starts at $a_1$ and ends at $a_i + a_j$. Likewise, there is no way to tell whether $a_i + a_j$ is supposed to occur at the $i$th index or the $j$th index or somewhere completely different.

So the primary purpose of writing out the $i-1$ and $i+1$ is for readability. By convention we accept this usage even in cases where it doesn’t quite match the template such as when $i=1,2,n-1,$ or $n$. The idea is that the reader can extract the pattern from the generic case and extrapolate it to the edge cases. In some cases it is really hard to convey the generic pattern using just “$\ldots$” ellipses (the plural of ellipsis), in which case adding more words usually helps :).