A = ( 0 1 ) B = ( 0 0 ) ( 0 0 ) ( 1 0 )acting on a two-component vector
v = ( yes ) ( no )Compute ABv and BAv. Suppose we apply this property to an access control system with two security checkpoints A and B. If we assign a matrix-valued security token after authentication to each secure area, how could we use this property to ensure that only those who have successfully be authenticated to A and then B (in that order) will gain access to data? ABv = yes, BAv = no. Obvious how we could use this? Give each user a matrix token/ticket on passing through a check point. Combine these with any matrix-tickets the use already has by multiplying the tickets at each step. To access a resource, we need to apply our ticket. If the result is yes, we get access otherwise no.
There are are many possible answers. Here is just one:
( 0 1 0 ) ( 0 0 0 ) ( 1 0 0 ) ( yes ) A = ( 0 0 0 ) B = ( 1 0 0 ) C = ( 0 0 0 ) and v = ( no ) ( 0 0 0 ) ( 0 0 0 ) ( 0 0 0 ) ( no )Can verify that only ABCv = yes. All other combinations give either "no" or zero for an answer. i.e. if we choose yes = 1, the this is consistent.