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Math Help - which is non singular?

  1. #1
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    which is non singular?

    Hello.
    I have two doubts
    Let A be a 3 x 3 matrix with eigen values 1,-1,3. then which is the correct option?
    a) A^2 + A is non-singular
    b) A^2 - A is non-singular
    c) A^2 + 3A is non-singular
    d) A^2 -3A is non-singular

    A is equivalent to the diagonal matrix having diagonal having diag entries 1,-1,3
    So A^2 s equivalent to the diagonal matrix having diagonal having diag entries 1,1,9.
    So I think the answer is c) A^2 + 3A is non-singular. Am I right??


    2) Let T : R -> R be a linear transformation and I be the identity transformation of R3. If there is a scalar c and a non zero vector x in R3 such that T(x)=cx, then the rank of (T-cI) cannot be
    a) 0
    b) 1
    C) 2
    D) 3

    c is the eigen calue of the transformation T. so T-cI is singular. So rank (T-cI) is not 3.
    Am i right??
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  2. #2
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    Quote Originally Posted by poorna View Post
    Hello.
    I have two doubts
    Let A be a 3 x 3 matrix with eigen values 1,-1,3. then which is the correct option?
    a) A^2 + A is non-singular
    b) A^2 - A is non-singular
    c) A^2 + 3A is non-singular
    d) A^2 -3A is non-singular

    A is equivalent to the diagonal matrix having diagonal having diag entries 1,-1,3
    So A^2 s equivalent to the diagonal matrix having diagonal having diag entries 1,1,9.
    So I think the answer is c) A^2 + 3A is non-singular. Am I right??


    Yes. You can also see this by checking the characteristic polynomial of A, p_A(x)=(x-1)(x+1)(x-3) This blows options (a),(b),(d) , so only (c) is left...



    2) Let T : R -> R


    According to the continuation of the question , this should be T\,:\,\mathbb{R}^3\to \mathbb{R}^3 , isn't it?


    be a linear transformation and I be the identity transformation of R3. If there is a scalar c and a non zero vector x in R3 such that T(x)=cx, then the rank of (T-cI) cannot be
    a) 0
    b) 1
    C) 2
    D) 3

    c is the eigen calue of the transformation T. so T-cI is singular. So rank (T-cI) is not 3.
    Am i right??
    Yes, you are.

    Tonio
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  3. #3
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    oh yeah, R3-> R3
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