In matrix multiplication, does ABC=ACB if A, B, and C are nxn matrices?

Question

17393 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-10-24T12:18:49

In general, no.

Matrix multiplication is associative but not commutative.

Let #A=( (1, 0), (0, 1) )#, #B=((1, 1), (0, 0))# and #C=((1, 0), (1, 0))#

Then:

#ABC = BC = ((1, 1), (0, 0))((1, 0), (1, 0)) = ((2, 0), (0, 0))#

#ACB = CB = ((1, 0), (1, 0))((1, 1), (0, 0)) = ((1, 1), (1, 1))#

What is true for all #n xx n# matrices #A#, #B# and #C# is:

#(AB)C = A(BC)#

That is, matrix multiplication is associative.