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For what values of \( n \) does there exist complex \( n \times n \) matrices \( A, B \) such that

\[ A B - BA = Id? \]

Note by Calvin Lin 2 years, 6 months ago

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If d=0, the condition holds trivially for all n by taking A=B=I. If d is different from zero, taking trace of both sides and using the fact that tr(AB)=tr(BA), we conclude that none such n exists.

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Id means identity matrix.

Anyways, I took it as identity matrix times the scalar d.

@Abhishek Sinha – No issues. That's a more general case :)

Notation for identity matrix isn't standardized anyway. I was debating between I and Id.

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`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestIf d=0, the condition holds trivially for all n by taking A=B=I. If d is different from zero, taking trace of both sides and using the fact that tr(AB)=tr(BA), we conclude that none such n exists.

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Id means identity matrix.

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Anyways, I took it as identity matrix times the scalar d.

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Notation for identity matrix isn't standardized anyway. I was debating between I and Id.

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