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Homework Help: Statement on matrix and determinant

  1. Apr 27, 2015 #1
    1. The problem statement, all variables and given/known data
    If A is a square matrix of order 3 then the true statement is
    1. det(-A) = - det A
    2.det A = 0
    3.det ( A + I) = I + detA
    4.det(2A) = 2detA
    2. Relevant equations
    NA

    3. The attempt at a solution
    2. option is obviously not true.
    Making a random matrix A and verifying properties 1. , 3. , 4. would be lengthy.
    What is the approach for this?
     
  2. jcsd
  3. Apr 27, 2015 #2

    Orodruin

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    Did you try simply using some basic rules for determinants? How would you write the determinant of a matrix in terms of its components?
     
  4. Apr 27, 2015 #3
    I know det(AT) = det(A)
    for equal size square matrices,
    det(A)(B) = det(A)det(B),
    What property should I apply as these I think are not applicable here?
     
  5. Apr 27, 2015 #4

    Orodruin

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    Can you write multiplication by a constant as a multiplication with a matrix? In that case, what matrix?
     
  6. Apr 27, 2015 #5
    Is it the identity matrix ## I ## ?
     
  7. Apr 27, 2015 #6

    PeroK

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    Why not take ##A = I##, and calculate the determinants required? Just to see what happens. You are allowed to try things out with specific matrices!
     
  8. Apr 27, 2015 #7

    Orodruin

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    Do you get 3A by multiplying A with the identity?
     
  9. Apr 27, 2015 #8
    No. We get A only.
     
  10. Apr 27, 2015 #9

    Orodruin

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    So how would you get 3A?
     
  11. Apr 28, 2015 #10
    By multiplying A with 3 or 3##I##
     
  12. Apr 28, 2015 #11

    Orodruin

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    Yes, try the latter and apply your determinant relations.
     
  13. Apr 28, 2015 #12
    But by that I am getting options 1 and 4 true.
    Among that one is supposed to be true.
     
  14. Apr 28, 2015 #13

    Orodruin

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    You are then doing it wrong. What is the determinant of 3I?
     
  15. Apr 28, 2015 #14
    27
     
  16. Apr 28, 2015 #15

    Orodruin

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    Yes, so what is then the determinant of 2A expressed in det(A)?
     
  17. Apr 28, 2015 #16
    8det(A).Option 4 rejected.
    Option 1 is okay.
    What about option 3 ?
     
  18. Apr 28, 2015 #17

    Ray Vickson

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    You tell us.
     
  19. Apr 28, 2015 #18
    But we can't apply the property there of
    det(A)det(B) = det(A)(B) as there is no use of it?
     
  20. Apr 28, 2015 #19

    Orodruin

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    Why dont you try insering a well chosen matrix and see what comes out? I suggest one which will make the determinants easy to compute.
     
  21. Apr 28, 2015 #20
    Oh, thanks that option 3 is also false.
    But I have verified that by taking any random matrix.
    Is there any general proof of that or we learn it in higher sections?
     
  22. Apr 28, 2015 #21

    Orodruin

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    To prove that something is false you only need a counter example.
     
  23. Apr 28, 2015 #22

    Ray Vickson

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    Right: not only is (3) false, it does not even have any meaning. The left-hand-side det(A+I) is a number, while the right-hand-side det(A) + I is a number plus a matrix---which does not exist in any reasonable way.
     
  24. Apr 28, 2015 #23

    Orodruin

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    I always took the I there to mean 1, precisely since it does not have any meaning otherwise ...
     
  25. Apr 28, 2015 #24

    Fredrik

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    In some contexts (especially in books on functional analysis) it's considered acceptable to write down equations of the type "operator = number" and sums of the type "operator + number". The number is then interpreted as that number times the identity operator. So (3) could be interpreted as ##\det(A+I)I=I+(\det A)I##, but it seems very unlikely that this is the intended interpretation. I'm inclined to go with Ray Vickson's interpretation first (the equality is nonsense), and Orodruin's interpretation second (the I on the right is supposed to be 1).
     
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