# 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....

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>