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rock_mit182

  • one year ago

A, B are matrices nxn; If A is NON-invertible matrix, A times B IS non-invertible as well. Please help me !

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  1. beginnersmind
    • one year ago
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    Can you use determinants?

  2. rock_mit182
    • one year ago
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    oh i see...

  3. rock_mit182
    • one year ago
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    if A INVERSE does not exit then |A| =0

  4. rock_mit182
    • one year ago
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    So there are two cases when: B inverse exist and B inverse does't exist, in other words: |B| =! 0 OR |B| = 0 in any case : |AB| = |A|*|B|

  5. rock_mit182
    • one year ago
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    WHICH alwas have to be zero given the condition of A

  6. rock_mit182
    • one year ago
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    right ?

  7. beginnersmind
    • one year ago
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    right

  8. beginnersmind
    • one year ago
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    last step is to point out det(AB) = 0 -> AB is non-invertible

  9. rock_mit182
    • one year ago
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    dude plz help me out with another question

  10. beginnersmind
    • one year ago
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    Sure

  11. rock_mit182
    • one year ago
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    show that A^2+2A = - I has inverse WHERE: A is nxn matrix; I is the identity matrix

  12. beginnersmind
    • one year ago
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    Hm, let me think.

  13. rock_mit182
    • one year ago
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    of course

  14. beginnersmind
    • one year ago
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    I guess you can do this one with determinants too :)

  15. rock_mit182
    • one year ago
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    i guess on this one i only can use propeties of matrix..

  16. beginnersmind
    • one year ago
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    Do we need to prove that A has an inverse or that A^2+2A does?

  17. beginnersmind
    • one year ago
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    Anyway, I need to go, so I'll leave with a short proof with determinants From A^2 + 2A = -I, we have det(A)*det(A) + 2det(A) + det(I) = 0 substituting u = det(A) u^2 + 2u + 1 = 0, or detA = 1 or detA = -1, so in either case A is invertible. I don't have a non-determinant proof, but here's a hint: If you take the matrix A^2+2A = -I its rank is n. A^2 + 2A = (A +2I)(A), and now you can use some linear algebra that both A and A+2I are rank n.

  18. beginnersmind
    • one year ago
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    u^2 + 2u + 1 = 0, or detA = 1 or detA = -1, CORRECTION u = -1, or detA = -1

  19. rock_mit182
    • one year ago
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    thanks dude

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