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Binary operation

  1. Oct 30, 2008 #1
    1. The problem statement, all variables and given/known data

    A binary operation is defined by:
    the set of 2 x 2 matrices with real entries under matrix multiplication.
    Find the identity element and the inverse.

    2. Relevant equations

    identity element: a[tex]\o[/tex]e=e[tex]\o[/tex]a=a

    inverse: a[tex]\o[/tex]b=b[tex]\o[/tex]a=e

    3. The attempt at a solution

    I think that the identity element is a 2 x 2 matrix of zeroes with 1's along the diagonal. but how do i find the inverse b?
    Thank you.
  2. jcsd
  3. Oct 30, 2008 #2


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    In general A will not have an inverse B such that AB = Identity. The brute force method is try multiplying a matrix with elements a,b,c,d with a matrix with elements e,f,g,h such that the resulting matrix is the identity. Find e,f,g,h in terms of a,b,c,d. The hint is that ad-bc is going to have to be non-zero for you to be able to solve this equation (because you have to divide by ad-bc)
  4. Oct 30, 2008 #3
    so if i do this, i get:
    why do i divide by ad-bc?
  5. Oct 30, 2008 #4


    Staff: Mentor

    'cause you're not done yet. You have to find the numbers e, f, g, and h in order to say what the inverse is for a given matrix A.

    Here are a couple of 2 x 2 matrices, one of which has an inverse and the other doesn't.

    A = [1 1; 0 2] (listed by rows)

    B = [1 2; 3 6] (ditto)
  6. Oct 31, 2008 #5
    ok i see what you mean, i found e, f, g, and h:

    Now, what if i had to find the identity element and inverse of a set of 2 x 2 matrices with non-zero determinant (under matrix multiplication).
    Does this mean that no such 'b' exists?
  7. Oct 31, 2008 #6


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    If the determinant is non-zero (i.e. ad-bc =/= 0) then you pick your matrix (e,f,g,h) just as you found in your post. Hence in the set of all matrices with non-zero determinant, the identity matrix is the identity, and inverses exist just as you found. It's only when ad-bc=0 that you can't find an inverse
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