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11 / 6 / 5 / ?

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  No. 714
>>721
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  No. 715

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  No. 721 >>722
>>714
pls less combinatorics problems thank you
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  No. 722 >>723
>>721
Why don't you like combinatorics?
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  No. 723
>>722
a math problem general should have some variety but besides that, combinatorics isn't a useful thing to study directly and its problems are ultimately better understood and solved in representation theory frameworks.
when you understand a problem well you are able to turn it into some kind of elementary computations/combinatorics, but this isn't an indication to study combinatorics specifically, than to study mathematics which is inherently meaningful and learn how to convert the abstract formulation

combinatorics really is a subset of other more important stuff, you get combinatorial tools in algebraic topology, number theory, algebraic geometry, homotopy. e.g. simplicial sets contain a lot of what you would ever do with graphs. in number theory you have trees coming from finite fields. in algebraic geometry you have combinatorics of the grasmannian, riemann surfaces, combinatorial ideals, matroids.
just learn other stuff, plain combinatorics is a fad driven by funding for "applied math" and computer science departments. serious combinatorialists (June Huh for example) end up studying homological algebra anyway
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  No. 776 >>777
Was posed to me recently:

Do there exist nxn matrices X, Y such that XY - YX = I_n? Provide an example or proof of the negative.
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  No. 777 >>778
>>776
Yep, it exists. For
X=A⋅B X=A \cdot B
with A,B
n⋅n n \cdot n
matrices
≠In \neq I_n

because the product between matrices isn't commutative.
So \( X \cdot Y-Y \cdot X = A \cdot B \cdot Y - Y \cdot A \cdot B = I_n)
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  No. 778
>>777
Edit: Yep, it exists. For
X=A⋅B X=A \cdot B
with A,B nxn matrices
≠In \neq I_n

because the product between matrices isn't commutative.
So
X⋅Y−Y⋅X=A⋅B⋅Y−Y⋅A⋅B=In X \cdot Y - Y \cdot X = A \cdot B \cdot Y- Y \cdot A \cdot B = I_n

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