Results 1 to 4 of 4

Math Help - Proving matrix addition theorem

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    7

    Proving matrix addition theorem

    Hi, I'm in first year university and I'm new to proofs, so I was wondering if anyone here can help me out.

    The questions ask to prove:

    1) A + (B + C) = (A + B) + c

    and

    2) For each m x n matrix A, there is unique m x n matrix D such that

    A + D = 0

    Any hints and insights would be greatly appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Apr 2009
    From
    United Kingdom
    Posts
    9
    Let's assume you're working over the real numbers.

    Let A = \left( {{a_{ij}}} \right), B = \left( {{b_{ij}}} \right) and C = \left( {{c_{ij}}} \right). Define D = A + (B + C) and E = (A + B) + C.

    By definition of the addition of matrices, we have {d_{ij}} = {a_{ij}} + ({b_{ij}} + {c_{ij}}) = ({a_{ij}} + {b_{ij}}) + {c_{ij}} = {e_{ij}}, since real numbers are commutative. So, D = E, and equivalently A + \left( {B + C} \right) = \left( {A + B} \right) + C.

    As for your second problem, letting A = \left( {{a_{ij}}} \right), then set D = \left( {{d_{ij}}} \right) = \left( { - {a_{ij}}} \right) (i.e. each entry of matrix D is the additive inverse of the same entry in A). Then it follows that A + D is equal to the zero matrix, and you're done.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2009
    From
    United Kingdom
    Posts
    9
    Futhermore, if you want to show that there is a UNIQUE matrix for 2), assume there are two different matrices which satisfy the condition, and work through all logical steps you can take - you'll get that they are equal to each other, and so your result is unique.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Sep 2009
    Posts
    7
    I think I understand now. Thanks for the help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Matrix multiplication/addition problem
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: April 11th 2011, 08:03 PM
  2. proving matrix theorem
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: March 11th 2011, 01:51 AM
  3. Proving that vector addition is associative
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: March 9th 2011, 02:57 PM
  4. proving an addition identity????
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: September 25th 2009, 02:28 PM
  5. Urgent Help: Matrix Addition For X And Y
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 21st 2006, 09:01 AM

Search Tags


/mathhelpforum @mathhelpforum