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
|
|
SOLUTION:
| 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.... |