Linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0

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  • #1
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linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0....

Homework Statement


A is a n*n matrix
A' is the matrix A when two two lines i,j are switched.
(switch two random lines is A and you get A')
If A Invertible Prove that the system (A+A')x=0 has infinite solutions

Homework Equations


linear algebra including Determinant


The Attempt at a Solution



Well I know that A+A' has two identical lines so when subtracting them I get a line of 0 and then Because I know there is a line of 0 I know that there are infinite solutions...
But I did not used the fact that A is Invertible...
How do I solve it while using this fact? , I try to use Determinant but I do not mange to.
Thank you.
 

Answers and Replies

  • #2
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let Det(A)=k
then Det(A')=-k (two rows are switched)

Det(A+A')=?
 
  • #3
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Thank you very much
 
  • #4
Dick
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let Det(A)=k
then Det(A')=-k (two rows are switched)

Det(A+A')=?
That logic doesn't work because det(A+A') is not equal to det(A)+det(A'). You need to think of something else ThankYou. Might it be that A+A' has two identical rows? How can you use that? You don't need the premise that A is invertible.
 
  • #5
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That logic doesn't work because det(A+A') is not equal to det(A)+det(A'). You need to think of something else ThankYou. Might it be that A+A' has two identical rows? How can you use that? You don't need the premise that A is invertible.
My bad.
 
  • #6
21
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Ha...
I've just got the Homework result back...
88
I got -6 for this question..
I know about the two identical lines but it did not used the fact that A is invertible...

Actually they just put the invertible fact to confuse because the teacher solution was the same as Dick without using the invertible Fact

Thanks anyway.
 

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