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Matrices/Linear Algebra Proof 
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#1
Sep810, 10:29 PM

P: 341

1. The problem statement, all variables and given/known data
Let A be an n x n matrix. Show that if A^{2}=0 then I  A is nonsingular and (I  A)^{1}= I + A 2. Relevant equations 3. The attempt at a solution Ok, so I started off with finding the general form of a 2x2 matrix which when squared gives a zero matrix, and all the properties above are satisfied. But how do I show that for an n x n matrix? Please help me out. 


#2
Sep810, 10:35 PM

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What is (IA)*(I+A)?



#3
Sep810, 10:36 PM

P: 341

The Identity matrix.



#4
Sep810, 10:39 PM

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Matrices/Linear Algebra Proof



#5
Sep810, 10:41 PM

P: 341

But I only know that because the question says to show it and the "experiment" I did with my 2x2 matrix shows that. I cannot prove it. Please give a more thorough explanation :)



#6
Sep810, 10:42 PM

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#7
Sep810, 10:44 PM

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Yes, obviously. I do not know how that is connected to the question though. It asks us to show that the inverse of IA is I+A which at this point is not selfevident (or at least I think so).



#8
Sep810, 10:49 PM

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#9
Sep810, 10:52 PM

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#10
Sep810, 10:55 PM

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#11
Sep810, 11:02 PM

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Can you do that with matrices? I have read the property that (A+B)C=AC+BC but only to that extent.



#12
Sep810, 11:04 PM

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#13
Sep810, 11:05 PM

P: 162

Yes, suppose C = D+E, hence AC+BC=(A+B)(D+E).



#14
Sep810, 11:08 PM

P: 341

Of course. That was kind of a stupid question. Thanks for clearing it up.
The question first asks us to show that IA is nonsingular. How do we show that? Or does that follow from (IA)(I+A)=I and since I has an inverse, it must be nonsingular? 


#15
Sep810, 11:10 PM

P: 162

When two matrices are multiplied together to produce the identity, it means the two matrices are inverses of each other. What is the definition of a nonsingular matrix?



#16
Sep810, 11:11 PM

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#17
Sep810, 11:13 PM

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A matrix multiplied by the original matrix to give Identity.



#18
Sep810, 11:18 PM

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