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$A$ and $B$ are two matrices, when is $(A-B)(A+B)=A^2 - B^2$

entrelac
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Hisham
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1 Answers1

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$$(A-B)(A+B)=A^2+AB-BA-B^2$$ So $(A-B)(A+B)=A^2-B^2$ is true when $AB=BA$ (matrix multiplication is commutative)

entrelac
  • 4,681
  • I think OP wants to know what specific properties a matrix should have so that $AB = BA$ – DeepSea Aug 23 '15 at 08:21
  • can you find the answer for this?Let A,B, and C be matrices of size l×m,m×n and n×p .In which order should be the triple product ABC be computed, so as to minimize the number of multiplication required? – Hisham Aug 23 '15 at 08:22
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    @Ganymede There aren't really any simple properties that are noth necessary and sufficient, except just to say that $AB = BA$. You could say "$A$ and $B$ are simultaneously diagonalizable", but that only works if you already know that $A$ and $B$ are diagonalizable, which isn't always. Of course, you could say things like "there is a $C$ such that $A = C^n$ and $B = C^m$", or "Either $A$ or $B$ are diagonal", but none of those are close to covering all cases either. – Arthur Aug 23 '15 at 08:23
  • @Ganymede I simply thought the OP might have not noticed that matrix multiplication is non commutative. And yes, I agree, the OP can refer to this post: http://math.stackexchange.com/questions/170241/when-is-matrix-multiplication-commutative – entrelac Aug 23 '15 at 08:25
  • @Hisham This is not homework forum. And not a place to ask many homework questions in one page. You need to give your attempts. If you do that again, you are likely to be banned and your questions will get downvoted and deleted forever. – làntèrn Aug 23 '15 at 08:26
  • @Hisham I saw your other post. You should try editing it a little bit and adding a few personal thoughts before asking for answers – entrelac Aug 23 '15 at 08:26