Proofs Help - Matrices

I need some help proving the Matrix Addition in general is associative. Any help is apriciated. Thank you

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Originally Posted by holmesb

I need some help proving the Matrix Addition in general is associative. Any help is apriciated. Thank you

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Let A B and C be NxM matrices and X_{i,j} denote the i,j th elements of a matrix C. As the i j th component of the sum of two matrices is the sum
of their individual i j th components and ordinary addition is associative:

Then [(A+B)+C]_{i,j} = (A+B)_{i,j} + C_{i,j} = A_{i,j} + B_{i,j} + C_{i,j}

.............................= A_{i,j} + (B+C)_{i,j} = [A+(B+C)]_{i,j}

RonL

Hello, holmesb!

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Prove that Matrix Addition is associative.

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To add two similar matrices, we add the corresponding elements and create a third matrix.
. . Symbolically, this can be wirrten: .P + Q .= .(pij) + (qij) .= .(pij + qij)


Given three similar matrices: .A, B, C
. . we want to show that: .A + (B + C) .= .(A + B) + C

Then: .A + (B + C) .= .(aij + [bij + cij]) .[1]

.and: .(A + B) + C .= .([aij + bij] + cij) .[2]


Since addition is associative, [1] = [2].

. . Therefore: .A + (B + C) .= .(A + B) + C

Thankyou both for your help.