Present this matrix as a multiplication of elementary matrices question

[transgalactic]

how to present this matrix as a multiplication of elementary matrices
$$\bigl(\begin{smallmatrix}<br /> 0 &6 &2 \\ <br /> 1& 1 &0 \\ <br /> 5&3 &1 <br /> \end{smallmatrix}\bigr)$$

i can't understand in general what are they doing in this solution
http://img144.imageshack.us/img144/9508/34036247.th.gif

??

 

[HallsofIvy]

I don't know what you mean by "I understand in general". Do you know how to "row-reduce" a matrix? Every "row-operation" corresponds to an elementary matrix: the matrix you get by performing that same row operation on the identity matrix.

There are many different ways to row-reduce any matrix but I like to start on the left and clear one column at a time. For example, I might, as the first step in row-reduction, swap the first two rows, going from
$$\begin{bmatrix} 0 & 6 & 2 \\ 1 & 1 & 0 \\ 5 & 3 & 1\end{bmatrix}$$
to
$$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 6 & 2 \\ 5 & 3 & 1\end{bmatrix}$$

Which corresponds to the elementary matrix we get by swapping the first two rows in the identity matrix:
$$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{bmatrix}$$

The next thing I might do is subtract 5 times the (new) first row from the third.
$$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 6 & 2 \\ 0 & -2 & 1\end{bmatrix}$$
and that corresponds to the elementary matrix
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -5 & 0 & 1\end{bmatrix}$$
What we have done so far corresponds to the product of those two elementary matrices. Continue to row-reduce and write down the corresponding elementary matrix. Be careful about the order of multiplication.

 

[transgalactic]

so when multiply all the elementary matrices
i get the original one

but they multiply only the first 4??

and i can't understand what's E what F in the equations
??

 

[HallsofIvy]

so when multiply all the elementary matrices
i get the original one

but they multiply only the first 4??

and i can't understand what's E what F in the equations
??

There are 6 elementary matrices. I don't see where they are multiplying "only the first 4".
The "E" matrices are the elementary matrices corresponding to the row operations needed to row-reduce matrix A. The "F" matrices are their inverses (which are also elementary matrices). A is the product of the "F" matrices.

 

[transgalactic]

i looked in a paper r and when there are operations like L1-5L2->L2
then on the elemtary matrix part
they say
L1+5L2->L2
i was told that its because they was to create elementary matrices which are invertable to itself X=X^-1
E1*E2*E3*...*A=I
so
E1^-1*E2^-1*E3^-1..*E1*E2*E3*...*A=I*E1^-1*E2^-1*E3^-1..
so
A=E1^-1*E2^-1*E3^-1..
but the inverses are the as the originals so we just need to multiply the
elementary matrices
correct??