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### Course: Algebra (all content)>Unit 20

Lesson 9: Properties of matrix multiplication

# Is matrix multiplication commutative?

Sal checks whether the commutative property applies for matrix multiplication. In other words, he checks whether for any two matrices A and B, A*B=B*A (the answer is NO, by the way). Created by Sal Khan.

## Want to join the conversation?

• In the 2x2 example, the products were different, but they have the same determinant. Is there any significance to this? Is it because one of the matrices is diagonal?
• good observation!
This is true en general; even though AB =/=BA we have the result:
Det(AB)=DetA · DetB
which is clearly symmetric. but note that A and B must be nxn and have Det =/= 0
• Are their other kinds of multiplication that are non-commuintive or is it just for matrices?
• Yes! Matrices are members of non commutative ring theory. Non commutative ring theory deals specifically with rings that are non-commutative with respect to multiplication. (For now lets just call a ring some algebraic structure - but I encourage you to do your own research!) These types of groups are not that well understood as are commutative rings. Examples would be Hamilton's quaternions, which are an extension of the complex numbers, and any algebraic group that is not abelian.
I encourage you to look up the following mathematical terms:
group, ring, abelian, quaternion, commutative and non-commutative (in the general sense)
to see how they are related.
Keep Studying!
• How would one go about reversing the A x B = C to find B given A and C?
• You can take the inverse of A and multiply C by that matrix:

A^(-1) x A x B = A^(-1) x C
I x B = A^(-1) x C
B = A^(-1) x C
• How would matrix multiplication apply to real life scenarios?
• Are there any special cases where matrix multiplication is commutative? Or does it not stand true for every matrix multiplication?
• Yes, if the matrices are SQUARE and identical. But that is pretty rare.
• I've had this thought for a while, when you add matrices , couldnt the mathematicians have said that ,for example
`` M=  [2,4,]          and  A = [2,3,3,]    [ 4,2]                   [ 3,2,3]``

cant you just pretend that the 3rd column is there but just full of zeros so you would get
`` M+A = [4,7,3,]       [ 7,4,3]``
?
• I think the asking member means is that if the mathematicens who had orginally defined the matrix concept in the convention could have thought that there was a unviersal set of null values. and that the matrices that were selected were only like sets in this universal matrix that unlike that rest of the realm have value other than null. So hence they these are selected and thus derive a matrix. Then in addition the matrices would simply be layered which would result in the odd result mentioned by the questioner.
• In what types of problems would it be helpful to know if matrix multiplication is commutative?
• Matrix multiplication is NOT commutative. The only sure examples I can think of where it is commutative is multiplying by the identity matrix, in which case B*I = I*B = B, or by the zero matrix, that is, 0*B = B*0 = 0.
There is a special case involving Simultaneous Diagonalization, and when both matrices are diagonal, but that is beyond this course.
• I noticed in the 2x2, that the ones that didn't match up to the -6 and 6 (4 and 9) were 2/3 and 3/2 (reciprocals) of the -6 and 6... Was that just coincidental, or is there any significance to that?
• This occurred only because the one matrix had zeros in it's antidiagonal. If non-zero values had been inserted in the antidiagonal, then the two commutative results would have no discernible correlation.

``    ┌       ┐    │  1  2 │A = │       │    │ -3 -4 │    └       ┘    ┌       ┐    │ -2  1 │B = │       │    │  1 -3 │    └       ┘      ┌      ┐      │ 0 -5 │A•B = │      │      │ 2  9 │      └      ┘      ┌       ┐      │ -5 -8 │B•A = │       │      │ 10 14 │      └       ┘``

http://en.wikipedia.org/wiki/Main_diagonal