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Thread: Proving matrix addition theorem

  1. #1
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    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!
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  2. #2
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    Let's assume you're working over the real numbers.

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

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

    As for your second problem, letting $\displaystyle A = \left( {{a_{ij}}} \right)$, then set $\displaystyle D = \left( {{d_{ij}}} \right) = \left( { - {a_{ij}}} \right)$ (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.
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  3. #3
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    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.
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  4. #4
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    I think I understand now. Thanks for the help!
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