I saw the Cayley-Hamilton theorem

Let $\mathbf{A}$ be a $n\times n$-matrix, and $p(\lambda)=\det(\lambda \mathbf{I}_n-\mathbf{A})$ the characteristic polynomial of $\mathbf{A}$. Then $p(\mathbf{A})=\mathbf{0}$

and I thought it was trivial since $\det(\mathbf{A}-\mathbf{A})=0$. I checked with Wikipedia and learned that this didn't work because $\lambda$ is a scalar, and $\det(\mathbf{0})$ is also a scalar, and not a matrix. This brings me to my question:

Are there any other results that seem trivial, but aren't?


marked as duplicate by user334732, Arnaud D., Community Apr 10 '18 at 9:52

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The Jordan curve theorem states that, if you draw a closed non-intersecting curve in the plane, it divides the plane into those points that are inside the curve, outside it or on it.


The weak Goldbach conjecture (a theorem now, it seems, proved by Harald Helfgott): Every odd integer $n>5$ is a sum of three primes. If you try yourself examples, this seems trivially true. There are millions of possibilities for a big odd $n$. Nevertheless, the proof is extremely complicated.

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    $\begingroup$ In case anyone doesn't know, it was proved in 2013 by Harald Helfgott. (I looked it up because I didn't realise it had even been proven.) $\endgroup$ – J.G. Apr 9 '18 at 16:47
  • $\begingroup$ Yes, there are $5$ arXiv papers, see wikipedia. However, an article with a proof has not been published in a journal yet. $\endgroup$ – Dietrich Burde Apr 9 '18 at 18:05

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