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So I am struggling with a couple very simple proofs of properties of matrix algebra. This is the first time I have ever had real proofs in math (Linear algebra). For the first one, I have it from our text but need a little help, and I am completely lost on the second one.

1) Prove that for matrices A and B that

A + B = B + A

We must show that the entries are identical for each. Therefore

(A+B)_{ij}= (B+A)_{ij}

= A_{ij}+ B_{ij}(Definition of addition??)

= B_{ij}+ A_{ij}(Commutativity of real numbers)

= (B+A)_{ij}(Definition of addition??)

So I totally get step 2. We're proving the commutativity of matrices, so we are allowed to use the commutativity of real numbers. But when the book says "definition of addition" it seems like they mean to say "distributivity".... so this is throwing me off.

Proof number 2 is:

Prove that (c+d)A = cA + dA where c,d are scalars and A is a matrix.

The only thing I can think to do is assume that their entries are equal, like in the first one, but then I am not sure where to go from there. So in summary,

1) Why does the book say "definition of addition" when it does and,

2) What is the first/second step for proof #2?

The book really doesn't provide any strategies for proofs, it seems like every proof is different at this point. I'm just not "seeing" it yet. Thanks so much in advance,

-Lee

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# Simple matrix algebra proofs

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