Suppose that: E_6 [[1,3,4],[2,2,3],[-5,4,4]] = [[1,3,4],[2,2,3],[-2,13,16]]   Find E_6.

Asked on

1 Answer | Add Yours

mathsworkmusic's profile pic

Posted on (Answer #1)

`E_6` is an elementary matrix. That is it is a matrix that can be reached from the identity matrix of the same dimension by an elementary operation.

We have that

`E_6 ((1,3,4),(2,2,3),(-5,4,4)) = ((1,3,4),(2,2,3),(-2,13,16))`

We can see that the first two rows are preserved and the third row is neither a multiple of itself nor a multiple of the other two rows. Eliminating these possibilities we are left with the fact that row 3 must be a linear combination of the rows.

Write `E_6 = ((1,0,0),(0,1,0),(a,b,c))`. Then

`((1,0,0),(0,1,0),(a,b,c))((1,3,4),(2,2,3),(-5,4,4)) = ((1,3,4),(2,2,3),(-2,13,16))`  so that

` ` `((1,2,-5),(3,2,4),(4,3,4))((a),(b),(c)) = ((-2),(13),(16))`

Reduce by row reduction:

`((1,2,-5),(0,-4,19),(0,-5,24))|((-2),(19),(24))` , ```((1,2,-5),(0,1,-19/4),(0,-5,24))|((-2),(-19/4),(24))`

`((1,2,-5),(0,1,-19/4),(0,0,24-95/4))|((-2),(-19/4),(24-95/4))` ,

`((1,2,-5),(0,1,-19/4),(0,0,1))|((-2),(-19/4),(1))` ,  `((1,2,0),(0,1,0),(0,0,1))|((3),(0),(1))` , `((1,0,0),(0,1,0),(0,0,1))|((3),(0),(1))`

Therefore `a=3` and `c=1` so that `E_6` is reached from the identity matrix by replacing row 3 by (3 x row 1) + row 3. We then have that

`E_6 = ((1,0,0),(0,1,0),(3,0,1))`

Answer. Note this could have been obtained by inspection but may have been difficult to spot as there were many possible combinations.


We’ve answered 397,000 questions. We can answer yours, too.

Ask a question