Hi, I need some help with proofs. Can someone help me with proving all the prope

Question

Hi, I need some help with proofs.

Can someone help me with proving all the properties of Matrix Multiplication without using Sigma notation? (using the i row and j column).
I am having trouble with proof type questions.
I just need it for studying for exam so its not a homework.

Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are defined.

a. A(BC) = (AB)C
b. A(B+C) = AB+AC
c. (B+C)A = BA+BC
d. r(AB) = r(A)B = A(rB) for any scalar r
e. Im A = A In = A

Thank you so much for your time I really really need some help so..please don't answer it if it has nothing to do with math... Thanks again Hi, I need some help with proofs.

Can someone help me with proving all the properties of Matrix Multiplication without using Sigma notation? (using the i row and j column).
I am having trouble with proof type questions.
I just need it for studying for exam so its not a homework.

Let A be an m x n matrix, and let B and C have sizes for which the indicated sums and products are defined.

a. A(BC) = (AB)C
b. A(B+C) = AB+AC
c. (B+C)A = BA+BC
d. r(AB) = r(A)B = A(rB) for any scalar r
e. Im A = A In = A

Thank you so much for your time I really really need some help so..please don't answer it if it has nothing to do with math... Thanks again

Explanation / Answer

Ans. Proofing without sigma notation is very much difficult. However in this case you can take a simple example, and then proof the relation. say m=2 & n=3, then choose an arbitary matrix. In this case arbitary matrix should be a general matrix containing variables not a fixed number.

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