How do I write this matrix as a product of elementary row 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 Do you do after you get the identity matrix?

What I am doing: So I first row reduce it to the identity matrix:

• I first switch $R_1$ and $R_3$

• Then $R_3-3R_1$

• then $R_2+2R_1$

• then $-1/3R_2$

• then $R_3-8R_2$

• then $R_2-2R_1$

• then $3/32R_3$

• then $R_1+3R_3$

• and finally $R_2+4/3R_3$

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

Since you are not showing you matrices, I am guessing the most likely mistake is

$$R_2-2R_1$$

Should be

$$R_2\color{red}+2R_1$$

Edit:

Most likely another mistake is $R_3-8R_1$ should be $R_3-8R_\color{red}2$

Edit:

After you perform row operations:

$$E_k\ldots E_1 A = I$$

$$A =E_1^{-1} \ldots E_k^{-1} $$