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I was solving a problem where you had to prove that some number $=0$. My strategy was to show that $Ak=A$ for some $k$ not equal to 1, hence $A(k-1)=0$ from which it follows that $A=0$.

Abstracting away from that particular problem I was solving (which I conveniently didn't tell you about), what other ways are there of proving that something $=0$?

Note: Other topics of this sort should be made, e.g. 'Ways of proving that $a\mid b$', etc.

Ahaan S. Rungta
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Ninja Boy
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    Good idea. I was thinking about the same thing. – user45220 Dec 15 '14 at 12:25
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    Every equation can be written is the form something=0. This is too broad for a question. – Marc van Leeuwen Dec 15 '14 at 12:25
  • You should be careful to specify in which structure you are working. For example $2 \times 4 = 2 \pmod{6}$. – quid Dec 15 '14 at 12:36
  • I do not really know what is part of precalculus algebra; but it seems from a page I just browsed that it contains matrices; see my comment on the answer how this can be relevant. Do you also cover congruences? If yes the comment here is relevant. Generally speaking that $ab=0$ imples that $a=0$ or $b=0$ is not true in all situations. It is true when working with real numbers, but not always. – quid Dec 15 '14 at 13:21
  • Sorry, no, I really don't know this, as high-school curricula vary quite a bit by country. But plenty of others on this site might know, so I will leave this to others. – quid Dec 15 '14 at 13:46

1 Answers1

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Here's three possible strategies.

Prove that $A\ge 0$ and $A\le 0$.

Prove that $A^2=0$.

Prove that $-A=A$.

paw88789
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  • You should be careful to specify in which structure you claim this to be true; the second is not true for the matrices, $(0, 1; 0,0)$. The third is not true in characteristic $2$. – quid Dec 15 '14 at 12:37