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Math Help - Inverse matrices=]

  1. #1
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    Inverse matrices=]

    Hi I have an assignment question as follows

    A, B and C are ntimes n matrices such that AB=I and CA=I
    show B= C (i have done this part)
    b) i. A and B are n times n matrices that commute. Show A squared and B squared commute
    ii. Give a generalisation of this result (without proof)
    c. A and B are n times n matrices and n is invetible. Shoe

    (A+B) A^-1(A-B)=(A-B)A^-1(A+B)

    d. A and B are n times n invertible matrices that commute. Show that A^-1 and B^-1 also commute

    it's fairly urgent-any help would be much appreciated
    thanks
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  2. #2
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    Quote Originally Posted by tkau8143 View Post
    Hi I have an assignment question as follows
    Since this is an assignment, you are expected to do it! Here are some hints.

    A, B and C are ntimes n matrices such that AB=I and CA=I
    show B= C (i have done this part)
    b) i. A and B are n times n matrices that commute. Show A squared and B squared commute
    ii. Give a generalisation of this result (without proof)
    I presume you have done this- it's almost trivial.

    c. A and B are n times n matrices and n is invetible. Shoe

    (A+B) A^-1(A-B)=(A-B)A^-1(A+B)
    I presume you mean "A is invertible".

    Go ahead an multiply out left and right sides. You should get the same result. The only difference between this and elementary algebra is that you have to be careful not to commute A and B.

    d. A and B are n times n invertible matrices that commute. Show that A^-1 and B^-1 also commute

    it's fairly urgent-any help would be much appreciated
    thanks
    This is also close to being trivial.

    Look at (AB)^{-1} and (BA)^{-1}. Of course since A and B commute, those must be equal.
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