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:
di,j = ∑k ai,k*bk,j
gi,j = ∑k bi,k*ck,j
fi,j = ∑k di,k*ck,j
(5) So, expanding fi,j gives us:
fi,j = ∑k (∑l ai,l*bl,j)ck,j =
(∑k ∑l) ai,l*bl,k*ck,j =
= ∑l ai,l*(∑k bl,k*ck,j) =
= ∑l ai,l*gl,j = hi,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:
di,j = ∑k ai,k*bk,j
(2) Let E = AC such that for each:
ei,j = ∑k ai,k*ck,j
(3) Let F = D + E = AB + AC such that for each:
fi,j = ∑k ai,k*bk,j+ai,k*ck,j = ∑k ai,k[bk,j + ck,j]
(4) Let G = B+C such that for each:
gi,j = bi,j + ci,j
(5) Let H = A(B+C) = AG such that for each:
hi,j = ∑k ai,k*gk,j
(6) Then we have AB + AC = A(B+C) since for each:
hi,j = ∑k ai,k[bk,j + ck,j]
(7) Let M = A + B such that for each:
mi,j = ai,j + bi,j
(8) Let N = (A+B)C = MC such that:
ni,j = ∑k mi,k*ck,j =
= ∑k (ai,k + bi,k)*ck,j
(9) Let O = BC such that:
oi,j = ∑k bi,k*ck,j
(10) Let P = AC + BC = E + O such that:
pi,j = ei,j + oi,j =
= ∑k ai,k*ck,j + ∑k bi,k*ck,j =
= ∑k [ai,k*ck,j + bi,k*ck,j] =
= ∑k (ai,k + bi,k)*ck,j
QED
Property 3: Scalar Multiplication
c(AB) = (cA)B = A(cB)
Proof:
(1) Let D = AB such that:
di,j = ∑k ai,k*bk,j
(2) Let E = c(AB) = cD such that for each:
ei,j = c*di,j = c*∑k ai,k*bk,j
(3) Let F = (cA)B such that:
fi,j = ∑k (c*ai,k)*bk,j = c*∑k ai,k*bk,j
(4) Let G = A(cB) such that:
gi,j = ∑k ai,k*(c*bk,j) =
= c*∑k ai,k*bk,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.
5 comments :
Hi,
I am a math teacher.
You can revise your understanding of matrices solving exercises about addition, subtraction and multiplication of matrices. Inverse,... at www.emathematics.net
Fantastic!!!
hi
i am a student taking a course in linear algebra. i just wanted to say that your tutorrial is very helpfull. so post some more on this topic.
thank you
Found it helpful. Thank you!
Another page that has a correct proof (avoiding the error noted above) can be found here:
http://www.proofwiki.org/wiki/Matrix_Multiplication_is_Associative#Proof
Hi Amittai,
Thanks very much for pointing out the mistake. I will revise the proof and post a comment on this blog when it is fixed.
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