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Math Help - Simple linear algebra question - opinion needed

  1. #1
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    Simple linear algebra question - opinion needed

    Good day to all.

    We have just completed our chapter on matrix operations (matrix matrix product, special matrices etc...). In doing the end of chapter exercises I have across a problem where the following is asked:

    Question: Find all 2x2 matrices where A^2 = 0

    I considered the general 2x2 matrix A = [[a,b], [c,d]] where [a,b] and [c,d] are the rows of A.

    Once I calculated AA I retrieved four equations which were the following:

    (1) a^2 +bc = 0 => bc = -(a^2) => c= -(a^2) / b with b <> 0
    (2) ab + bd = 0 => b(a+d) = 0 => b=0 or a = -d
    (3) ac + dc = 0 => c(a+d) = 0 => c=0 or a = -d
    (4) bc + d^2 = 0 => bc = -(d^2)

    From here I assumed that a = -d and constructed the following matrix:

    A = [[a,b],[(-a^2 / b), -a]] where the inner brackets represent the rows of A.

    I was just wondering if the process used is correct? I apologize if the presentation leaves a lot to be desired. I am not sure of how to code matrices. Any help would be greatly appreciated.

    Kindest regards
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  2. #2
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    Quote Originally Posted by gate13 View Post
    Good day to all.

    We have just completed our chapter on matrix operations (matrix matrix product, special matrices etc...). In doing the end of chapter exercises I have across a problem where the following is asked:

    Question: Find all 2x2 matrices where A^2 = 0

    I considered the general 2x2 matrix A = [[a,b], [c,d]] where [a,b] and [c,d] are the rows of A.

    Once I calculated AA I retrieved four equations which were the following:

    (1) a^2 +bc = 0 => bc = -(a^2) => c= -(a^2) / b with b <> 0
    (2) ab + bd = 0 => b(a+d) = 0 => b=0 or a = -d
    (3) ac + dc = 0 => c(a+d) = 0 => c=0 or a = -d
    (4) bc + d^2 = 0 => bc = -(d^2)

    From here I assumed that a = -d and constructed the following matrix:

    A = [[a,b],[(-a^2 / b), -a]] where the inner brackets represent the rows of A.

    I was just wondering if the process used is correct? I apologize if the presentation leaves a lot to be desired. I am not sure of how to code matrices. Any help would be greatly appreciated.

    Kindest regards

    Looks fine , but...what if b=0 ? It's just a simple particular case, but I think you should cover it.

    Tonio
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