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Math Help - matrix multiplication commutativity

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    matrix multiplication commutativity

    Let A & B be matrices such that:

    A.B = B.A (with det(B) =/= 0)

    Show that B can be written under the form:

    B = mA + pI where m & p are real numbers and I is the identity matrix.
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    Quote Originally Posted by tombrownington View Post
    Let A & B be matrices such that:

    A.B = B.A (with det(B) =/= 0)

    Show that B can be written under the form:

    B = mA + pI where m & p are real numbers and I is the identity matrix.
    The two matrices that A always commutes with are itself and I.

    If A commutes with B it therefore follows that B must be a linear combination of A and I.
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    Sorry Mr. Fantastic, I didn't quite get you.
    Can you elaborate a little.
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    Quote Originally Posted by tombrownington View Post
    Sorry Mr. Fantastic, I didn't quite get you.
    Can you elaborate a little.
    Please be specific - what don't you get?
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    Quote Originally Posted by mr fantastic View Post
    The two matrices that A always commutes with are itself and I.

    If A commutes with B it therefore follows that B must be a linear combination of A and I.
    I don't follow the "it therefore follows that B must be a linear combination of A and I" part. That was the whole point of the question in any case.
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    Quote Originally Posted by tombrownington View Post
    I don't follow the "it therefore follows that B must be a linear combination of A and I" part. That was the whole point of the question in any case.
    You need to make the observation that A commutes only with itself (that is, with A) and I. Once this has been noted, it clearly follows that if A commutes with B then B must be a linear combination of the matrices that commute with A, that is, a linear combination of the matrices A and I.
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    Quote Originally Posted by tombrownington View Post
    Let A & B be matrices such that:

    A.B = B.A (with det(B) =/= 0)

    Show that B can be written under the form:

    B = mA + pI where m & p are real numbers and I is the identity matrix.
    This result is not true. For example, suppose that A is the identity matrix and B is any invertible matrix that is not a scalar multiple of the identity.
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    Quote Originally Posted by Opalg View Post
    This result is not true. For example, suppose that A is the identity matrix and B is any invertible matrix that is not a scalar multiple of the identity.
    Good point!
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