What could be added as context? One thing that comes to mind is saying that this is true for product - and the question is whether the same is true for sum?
improve I have also tried to edit Minimize $a+b+c+d$ given $\text{LCM}(a,b,c,d)=1000$ - this question cause some heated debate recently. I wonder whether the version after my edit would already be considered as having sufficient context.
I have edited the post (revisions, the version after my edit) in order to add at least some context. I have not case my undelete vote - I'll wait first to hear whether you are satisfied with the edit and leave some time for you if you want to edit it further. — Martin Sleziak3 hours ago
@MartinSleziak I see I did not read the question carefully enough. The "multiplicative analogue" would be false. (Product of an invertible matrix and a singular matrix is singular.)
@MartinSleziak Perhaps asking exactly that question might be the way to go. Make it a more pedagogical question?
> Suppose that $A$ is a singular matrix and that $B$ is nonsingular, where both are square of the same dimension. It is not hard to see that $AB$ and $BA$ are both singular. What about addition of matrices? Is $A+B$ always singular?
perhaps?
I still don't think that it is great, but it adds some context...
That being said, I think that the question itself has such an obvious answer if one takes more than two seconds to think about it that it probably doesn't need an answer here---asking the question is kind of worth a downvote for "not researching."
I feel like the right question is something somewhat more general about which properties of operators are preserved by what kinds of transformations... but I don't know quite what that question is.
@XanderHenderson In the case of the question 1993009 this sounds like a fair point. I have edited the question a bit anyway - and I'll leave the rest up to other users.
@MartinSleziak Thank you, again, for putting in the effort. I'm not going to vote to undelete, but the question is improved enough now that I wouldn't vote to redelete were it to be undeleted, either.
However, I'll also add that I had students that had problems coming up with an example of two bijections $\mathbb R\to\mathbb R$ such that the sum is not a bijection. So even things which might look obvious can make a beginner stumble sometimes. (Perhaps it depends on what way are they used to think about the objects which are discussed.)
In the case of adding matrices, however, the sum of a generic singular matrix with a generic nonsingular matrix will be nonsingular. One actually has to work to get examples where the sum is singular.
The example of bijective functions is a littlle bit more subtle, I think---I am not even sure that an "obvious" example of a bijection is to a beginning student (the identity function, maybe?).
I think that I could make a case that it isn't really about mathematics ("why" a thing happens in mathematics is more a question of philosophy---the best mathematical answer that I can think of is "because it does").
@XanderHenderson At least it's asked in curiosity. And technically there is a 'real' answer: it is an instance of the strong law of small numbers. It is not just 'luck', but in fact we expect to encounter good approximations now and then, and we can estimate the frequency of such occurrences based on the precision of the approximation relative to the bits needed to express the approximation.
But at the level of the asker, I think it's enough to say that its continued fraction has a large term after the truncation, as GEdgar said, which even though logically circular seems to me to convey roughly the idea of just how coincidental it is, which is slightly more satisfying than just saying it's coincidence.
$$e^\frac{1}{e} = \frac{13.002010749 \cdots}{9}$$
Is there a reason why it is close to $\frac{13}{9}$?
Or is this just a mathematical coincidence?
