Does the commutative property hold for the product of matrices?

Question

Does A X B = B X A in all cases? if no explain why with example

Answer 1

No!
Considering a matrix A (a1,1 a1,2) and a matrix
(a2,1 a2,2)
B (b1,1 b1,2)
(b2,1 b2,2)

, where the numbers 1,1; 1,2 etc. indicate in order the line and the clumn of the element in the matrix, to obtain a matrix C=AxB we have to operate as follows


c1,1 c1,2
| |
C (a1,1xb1,1+a1,2xb2,1 a1,1xb1,2+a1,2xb2,2)
(a2,1xb1,1+a2,2xb2,1 a2,1xb1,2+a2,2xb2,2)
| |
c2,1 c2,2

following this scheme you'll understand that you'll have the same result just if the matrix is sqared (= elements in the columns are as many as they are in the lines).

The proceding path is the same for all sizes matrices, of course the result changes if one matrix is sqared and the other is not.

hope i was helpful

Answer 2

Yes, it does.

Sue