Properties of Matrix Multiplication

Multiplication can only occur between matrices A and B if the number of columns in A match the number of rows in B. This presents the very important idea that while multiplication of A with B might be a perfectly good operation; this does not guarantee that multiplication of B with A is a perfectly good operation.

Even if matrix A can be multiplied with matrix B and matrix B can be multiplied to matrix A, this doesn't necessarily give us that AB=BA. In other words, unlike the integers, matrices are noncommutative.

Property 1: Associative Property of Multiplication

A(BC) = (AB)C

where A,B, and C are matrices of scalar values.

Proof:

(1) Let D = AB, G = BC
(2) Let F = (AB)C = DC
(3) Let H = A(BC) = AG
(4) Using Definition 1, here, we have for each D,F,G,H:

d_i,j = ∑_k a_i,k*b_k,j

g_i,j = ∑_k b_i,k*c_k,j

f_i,j = ∑_k d_i,k*c_k,j

(5) So, expanding f_i,j gives us:

f_i,j = ∑_k (∑_l a_i,l*b_l,j)c_k,j =

(∑_k ∑_l) a_i,l*b_l,k*c_k,j =

= ∑_l a_i,l*(∑_k b_l,k*c_k,j) =

= ∑_l a_i,l*g_l,j = h_i,j

QED

Property 2: Distributive Property of Multiplication

A(B + C) = AB + AC
(A + B)C = AC + BC

where A,B,C are matrices of scalar values.

Proof:

(1) Let D = AB such that for each:

d_i,j = ∑_k a_i,k*b_k,j

(2) Let E = AC such that for each:

e_i,j = ∑_k a_i,k*c_k,j

(3) Let F = D + E = AB + AC such that for each:

f_i,j = ∑_k a_i,k*b_k,j+a_i,k*c_k,j = ∑_k a_i,k[b_k,j + c_k,j]

(4) Let G = B+C such that for each:

g_i,j = b_i,j + c_i,j

(5) Let H = A(B+C) = AG such that for each:

h_i,j = ∑_k a_i,k*g_k,j

(6) Then we have AB + AC = A(B+C) since for each:

h_i,j = ∑_k a_i,k[b_k,j + c_k,j]

(7) Let M = A + B such that for each:

m_i,j = a_i,j + b_i,j

(8) Let N = (A+B)C = MC such that:

n_i,j = ∑_k m_i,k*c_k,j =

= ∑_k (a_i,k + b_i,k)*c_k,j

(9) Let O = BC such that:

o_i,j = ∑_k b_i,k*c_k,j

(10) Let P = AC + BC = E + O such that:

p_i,j = e_i,j + o_i,j =

= ∑_k a_i,k*c_k,j + ∑_k b_i,k*c_k,j =

= ∑_k [a_i,k*c_k,j + b_i,k*c_k,j] =

= ∑_k (a_i,k + b_i,k)*c_k,j

QED

Property 3: Scalar Multiplication

c(AB) = (cA)B = A(cB)

Proof:

(1) Let D = AB such that:

d_i,j = ∑_k a_i,k*b_k,j

(2) Let E = c(AB) = cD such that for each:

e_i,j = c*d_i,j = c*∑_k a_i,k*b_k,j

(3) Let F = (cA)B such that:

f_i,j = ∑_k (c*a_i,k)*b_k,j = c*∑_k a_i,k*b_k,j

(4) Let G = A(cB) such that:

g_i,j = ∑_k a_i,k*(c*b_k,j) =

= c*∑_k a_i,k*b_k,j

QED

Property 4: Muliplication of Matrices is not Commutative

AB does not have to = BA

Proof:

(1) Let A =



(2) Let B =



(3) AB =



(4) BA =



QED

References

• Hans Schneider, George Philip Barker, Matrices and Linear Algebra, 1989.