Tuesday, April 17, 2007

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:

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 =

(∑kl) 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

5 comments :

Eva Acosta said...

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!!!

Anonymous said...

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

Luis said...

Found it helpful. Thank you!

Anonymous said...

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

Larry Freeman said...

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.