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Math Help - A couple of Matrix Algebra Problems

  1. #1
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    Smile A couple of Matrix Algebra Problems

    Prove the following. In each case clearly state in mathematical notation what is given and what you are asked to prove.
    a) Given that A and B are square matrices that commute and satisfies
    A2 − AB − 2B2 − I = 0
    Prove that the inverse of (A+B) exists and find it.
    Hint: assume the inverse is of the form (αA + βB) or some constants α and β.

    b) Given that A , B and C are symmetric square matrices of the same size, prove that the matrix AT + B + CT is symmetric.

    AND...

    What is the rank of an invertible 5x5 matrix? Why?

    Thanks a lot. Any suggestions would be appreciated.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Nitz456 View Post
    What is the rank of an invertible 5x5 matrix? Why?
    the rank of an nxn matrix A is equal to the number of non-zero rows A has when it is put in reduced row-echelon form. since A is invertible, it means its reduced row echelon form is the identity matrix I_n. Thus, the reduced row-echelon form of a 5x5 matrix is I_5 which means the rank for such a matrix is 5
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