Question

If ABx=BAx, then is necessary that BA=AB?
(A and B are matrices and x is a column.)
If not then what are the cases in which AB=BA?

Answer 1

Do you mean for all x or just specific x? If you mean all x then I think you have to have AB=BA, because it would have to hold for all of x= (1,0,0,...,0), (0,1,0,...,0), ..., (0,0,0,...,1). Each of these vectors extracts a single column from the matrix to its left (AB or BA); if all the columns match then the matrices are equal. If on the hand you mean specific x then there is no requirement for AB= BA: even disregarding the x=0 case you can see that if say x=(1,0,0,...,0) then all that is required is that the first columns match.

Answer 2

AB = BA only if AB is invertible.

Answer 3

The original question was about whether ABx=BAx impllied that AB=BA, so I'm going a bit off topic here but I don't agree with bodhibrata's statement that AB=BA only if AB is invertible. Try A=[1 -1 1; -2 2 0; -2 2 0] and B=[2 -1 1; 4 -3 3; 4 -4 4]; then AB=BA=[2 -2 2; 4 -4 4; 4 -4 4], which is not invertible.

Answer 4

Yes u r right JoshDanziger23. I was wrong. Thanx

Answer 5

Thanks bodhibrata!