elementary matrix - single element change

Question

I have a question on how to get matrix B from matrix A.

A =   1 2 3 4 5 
      6 7 8 9 10
      0 0 1 0 0

B is given, too

B = 6 7 0 9 10
    1 2 0 4 5
    0 0 1 0 0

the tools are swapping, adding and multiplying individual lines to one another. main problem here: how to get these 0s in line 1 and 2 is not obvious for me in B.

Answer 1

First swap the first and second rows to get $\begin{pmatrix}6 & 7 & 8 & 9 & 10 \\ 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 1 & 0 & 0 \end{pmatrix}$

Then subtract 8 times the third row from the first row and subtract 3 times the third row from the second row.

Answer 2

Use the 3rd row to clear the 3rd column except the bottom entry then swap the 1st & 2nd row, period.

Answer 3

If the valid operations are 'swapping, adding and multiplying individual lines to one another', then it is impossible to get the matrix $B$ from matrix $A$. note that these operations can only increase the total sum of all the elements.

However, these are not the only operations allowed by multiplication by elementary matrices. Elementary transformation allow you to add to a row, another row multiplied by a number. This operation would allow you to add to the first row, the last row multiplied by $-1$. This is, subtract the last row from the first row. Likewise, you can subtract the last row from the second row. Finally swap the first two rows to get $B$.