# Show a matrix as a product of three elementary matrices

I am really not sure what I am doing wrong here so i'll show my steps and can someone please show me what I am doing wrong.

I need to express matrix A as a product of three elementary matrices

$$A=\begin{bmatrix}1 & 3\\1 & 2\end{bmatrix}$$

So this is how I attempted it:

Step 1: $R2 - R1$

$$E_1=\begin{bmatrix}1 & 0\\-1 & 1\end{bmatrix}$$ $$E_1A=\begin{bmatrix}1 & 3\\0 & -1\end{bmatrix}$$

Step 2: $R2 * -1$

$$E_2=\begin{bmatrix}1 & 0\\1 & -1\end{bmatrix}$$ $$E_2E_1A=\begin{bmatrix}1 & 3\\0 & 1\end{bmatrix}$$

Step 3: $R1 - (3*R2)$

$$E_3=\begin{bmatrix}-2 & 3\\1 & -1\end{bmatrix}$$ $$E_3E_2E_1A=\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$

Step 4: Check

$$E_3E_2E_1A = I_{2x2}$$ $$E_1^{-1}E_2^{-1}E_3^{-1}=A$$

Step 4 is where it all goes wrong and it doesn't actually add up. I've run through it a couple of times now and I think i'm missing something really simple so some insight would be great please.

• The thing you typed as E_3 is not an elementary matrix, it is a mess. Also, E_2 is incorrect! Oct 30 '17 at 18:40