if A and B are nxn matrices, then (A+B)(A-B)=A^2+B^2

I think this is true but I'm not sure why, but I tried it out with numbers and I can't get it to come out the same, but I think my numeric andwers are wrong..

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- November 3rd 2009, 04:16 AMcottekrtrue/false question
if A and B are nxn matrices, then (A+B)(A-B)=A^2+B^2

I think this is true but I'm not sure why, but I tried it out with numbers and I can't get it to come out the same, but I think my numeric andwers are wrong.. - November 3rd 2009, 04:59 AMHallsofIvy
I am puzzled at why you are doing this at all. Most people don't work with matrices until long after they have taken basic algebra and one of the things you should have learned there is that, for numbers, (A+ B)(A- B)= A^2- B^2. So I don't see why you would be "trying" numbers- you should know that it will never work.

Are you sure you have copied the problem correctly? The question of whether (A+B)(A-B)= A^2- B^2 would be much more interesting.

For either problem, the first thing you should have thought about was expanding the product: (A+B)(A-B)= A(A+B)+ B(A-B)= A(A)+ A(-B)+ BA+ B(-B)= A^2- AB+ BA- B^2. In order for that to be "A^2+ B^2", it must be true that -AB+ BA= 2B^2. Do you think it is at all likely that that will be true for**all**A? In order for that to be "A^2- B^2", It must be true that -AB+ AB= 0 which is the same as saying that AB= BA. Is that true for matrices?