Why Might the Square of a Sum Not Equal the Sum of Squares for Matrices?

  • Thread starter cmab
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In summary, the equation (e+f)^2 = e^2+2ef+f^2 is not necessarily true for square matrices of all sizes due to the non-commutativity of matrix multiplication. This was clarified in the conversation where it was pointed out that the correct equation is (e+f)^2 = e^2 + ef + fe + f^2.
  • #1
cmab
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0
(e+f)^2 = e^2+2ef+f^2
Why is this not neccesary true for squares matrices for all size?
 
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  • #2
(e+ f)^2= (e+f)(e+f)= e^2+ fe+ef+ fe. Remember that multiplication of matrices is not, in general, commutative!
 
  • #3
HallsofIvy said:
(e+ f)^2= (e+f)(e+f)= e^2+ fe+ef+ fe. Remember that multiplication of matrices is not, in general, commutative!
Do you mean (e+f)2 = e2+fe+ef+f2? I might be misinterpreting it but I can't see where that extra "fe" came from and where the "f2" went.
 
  • #4
Yeah, that's what he meant: (e+f)² = e² + ef + fe + f².
 
  • #5
One of these days, I'm going to learn to type!
 

FAQ: Why Might the Square of a Sum Not Equal the Sum of Squares for Matrices?

Why can this be false?

1. What factors can contribute to a false result in scientific experiments?

There can be various factors that can lead to a false result in scientific experiments. These can include human error, equipment malfunction, contamination, or insufficient sample size.

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2. How can we ensure the accuracy of scientific findings?

To ensure accuracy, scientists use various methods such as peer review, replication of experiments, and statistical analysis to validate their findings. They also follow strict protocols and guidelines to minimize the potential for error.

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3. Can scientific theories ever be proven false?

Yes, scientific theories can be disproven if new evidence or data contradicts them. However, this does not mean that the entire concept of science is false. It simply means that our understanding and knowledge of a particular phenomenon may have been incomplete or incorrect.

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4. Are there any limitations to the scientific method?

Yes, the scientific method has its limitations. It relies on observable and measurable evidence, so it cannot be used to study abstract concepts such as emotions or morality. Additionally, it is constantly evolving and subject to human bias and error.

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5. What is the role of skepticism in the scientific process?

Skepticism is a fundamental aspect of the scientific process. Scientists are trained to question and critically evaluate evidence and arguments to ensure the validity and reliability of their findings. It helps to prevent false conclusions and promotes the pursuit of truth.

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