Problem11:Consider the matrices, 
> ;
[3 4 1] [ ] A := [2 7 1] [ ] [8 1 5]> ;
[3 4 1] [ ] B := [2 7 1] [ ] [2 7 3]
Find elementary matrices E and F such that, 
> ;
E A = B F B = A

We only need to identify the elementary row operation that takes A into B. By inspection we see that B is obtained from A by adding (2) times the first row to the third row. The matrix representation E of an elementary row operation is obtained by applying the corresponding transformation to the identity matrix since, EI=E. Hence, 
> E := addrow( diag(1,1,1), 1,3,2);
[ 1 0 0] [ ] E := [ 0 1 0] [ ] [2 0 1]
let's check, 
> A := matrix(3,3,[3,4,1, 2,7,1, 8,1,5]):
> evalm( E &* A );
[3 4 1] [ ] [2 7 1] [ ] [2 7 3]
so yep! we got B. It works. Now let's do the trick in reverse, i.e. go from B to A with an elementary row operation. Clearly we get A when we add to the third row of B, (2)times its first row. So we have, 
> F := addrow( diag(1,1,1), 1,3, 2);
[1 0 0] [ ] F := [0 1 0] [ ] [2 0 1]
Now notice that F is the inverse of E as it should, right? 
> evalm( E &* F );
[1 0 0] [ ] [0 1 0] [ ] [0 0 1]
Did you make sure you understand the maple commands in this problem? Check the maple help on: addrow, diag, &* and evalm.... 