Answer
We use the definition of matrix addition and multiplication. All summations here are from 1 to k .
a) (A + B)C =[Σ (aiq + biq)cqj]
= [Σ aiqcqj + Σ biqcqj]
= AC + BC
b) C(A + B) = [Σ ciq(aqj + bqj )]
= [Σ ciqaqj + Σ ciqbqj]
= CA + CB
Work Step by Step
(a)
= ∑(A_ik + B_ik) * C_kj (using matrix addition)
= ∑(A_ik * C_kj + B_ik * C_kj) (distributive property)
= ∑(A_ik * C_kj) + ∑(B_ik * C_kj)
= AC_ij + BC_ij (distributive property and linearity of summation)
(b)
= ∑C_ik * (A_kj + B_kj) (using matrix addition)
= ∑(C_ik * A_kj + C_ik * B_kj) (distributive property)
= ∑C_ik * A_kj + ∑C_ik * B_kj
= CA_ij + CB_ij (distributive property and linearity of summation)
