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

  1. Jul 31, 2010 #1
    linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0....

    1. The problem statement, all variables and given/known data
    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

    2. Relevant equations
    linear algebra including Determinant


    3. 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.
     
  2. jcsd
  3. Jul 31, 2010 #2
    Re: linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0.

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

    Det(A+A')=?
     
  4. Jul 31, 2010 #3
    Re: linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0.

    Thank you very much
     
  5. Jul 31, 2010 #4

    Dick

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

    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.
     
  6. Aug 1, 2010 #5
    Re: linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0.

    My bad.
     
  7. Aug 13, 2010 #6
    Re: linear algebra A' is A when two of A lines switched,A Invertible prove (A+A')x=0.

    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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