Proof of the "Left Distributive Law" for matrices. ******A(B + C) = AB + AC****** Again we show that the general element of the left hand side is the same as the right hand side. We have (A(B + C))ij = S(Aik(B + C)kj) definition of matrix multiplication = S(Aik(Bkj + Ckj)) definition of matrix addition = S(AikBkj + AikCkj) distributive property of the real numbers = S AikBkj + S AikCkj commutative property of the real numbers = (AB)ij + (AC)ij definition of matrix multiplication where the sum is taken from 1 to k. S is the Sigma sign. ______________________________________________________________ Can somebody tell me why there is a sigma sign in the first step? Wouldn't the proof be correct without the sigma signs? How do they even derive the sigma or sums from the definition of matrix multiplication? Say we have a matrix A, and a matrix B, and say AB is defined, then the element in the first row and the first column can be computed by multiplying and adding the corresponding entries in the first row of A, and the first column of B. Now, for instance, k=1, then wouldn't AikBkj become (Ai1)(B1j), would simply mean the first column from A multplied by the corresponding entries from column 1 of B!!!!! But what is that supposed to mean? Shouldn't it have been the first row of A, and first column of B? Please help as soon as possible.