# How to write a matrix as a product of elementary matrices? [duplicate]

\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

## marked as duplicate by Dietrich Burde, Lord Shark the Unknown, user228113, Siong Thye Goh, projectilemotionNov 8 '17 at 19:50

• Please do not repost the same question! – Dietrich Burde Nov 8 '17 at 19:44
• Edit your previous post to include your working. We know you made a mistake. Show us your working and we can spot the mistake. – Siong Thye Goh Nov 8 '17 at 19:47
• Ok I will add my work now, sorry I am new tot this site. – G Muf Nov 8 '17 at 19:48
• The determinant is $32$, it is indeed nonsingular. – Siong Thye Goh Nov 8 '17 at 19:49
• Yes, and therefore it can be written as a product of elementary matrices. I just can't seem to do it. – G Muf Nov 8 '17 at 19:55

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.

Guide:

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

$$E_k \ldots E_1 A = I$$

Solve for $A$.