How to write a matrix as a product of elementary matrices?

Question

\begin{pmatrix}3&-1&3\\ -2&3&2\\ 1&-3&1\end{pmatrix}

Any help would be greatly appreciated! I have done this before with other matrices, yet this one is bringing me trouble. I have spent too long on this as I believe it should be easy and yet my answers are always wrong.

What I am doing: So I first row reduce it to the identity matrix: I first switch R1 and R3 Then R3-3R1 then R2-2R1 then -1/3R2 then R3-8R1 then R2-2R1 then 3/32R3 then R1+3R3 and finally R2+4/3R3

I know there is many possible ways to get this too the identity matrix but this worked for me. I then apply all of the above steps to identity matrices and multiply them together to check my work but it never works out to be the original matrix so I am forced to believe something is wrong

Answer 1

HINT

1. Apply Gauss-Jordan elimination reducing to identity
2. Express each step in Gauss-Jordan elimination a multiplication by an elementary matrix.
3. Invert the matrices and multiply in reverse order.

Answer 2

Guide:

Reduce the matrix to Reduced Row echelon form using elementary operations.

$$E_k \ldots E_1 A = I$$

Solve for $A$.