Write a proof for the following Properties of Matrix Multiplication. Let A, B, C be matrices.?

Write a proof for these Properties of Matrix Multiplication: Let A, B, C be matrices.





1. A(B + C) = AB + AC, whenever all products are defined.





2. A(cB) = (cA)B = c(AB), for any scalar c.

Write a proof for the following Properties of Matrix Multiplication. Let A, B, C be matrices.?
1) Assume all the matrix dimensions are n x n. Let a_ij be the entry in the ith row and jth column of A, b_ij for matrix B, etc. Then the i,j entry of B+C is (b_ij + c_ij). The i,j entry of the product is going to be the product of the ith row of A and the jth column of B+C. This is


[a_i1 (b_1j + c_1j)] + [a_i2 (b_2j + c_2j)] + [a_i3 (b_3j + c_3j)] + ... + [a_in (b_nj + c_nj)].





You'll see that you can rewrite this as:


[ a_i1 b_1j + a_i2 b_2j + a_i3 b_3j + ... + a_in b_nj ] +


[ a_i1 c_1j + a_i2 c_2j + a_i3 c_3j + ... + a_in c_nj ]





which is the same thing as


[ the i,j entry of AB ] + [ the i,j entry of AC ]





You can use a similar approach to prove #2, which should be easier.