linear algebra - Give the corresponding elementary matrix decomposition of A

can you guys explain the question to me

Put the following matrices into reduced row echelon form, indicating the row operations you use. Give the corresponding elementary matrix decomposition of A

$$\left[ \begin{array}{ccc} 2&1&1\\ 1&2&1\\ 1&1&2 \end{array} \right]$$

i put the matrix in RREF form, but i dont know how to get the elementary matrix.

Answer

Whenever you perform elementary row operations, you are multiplying the matrix by an elementary matrix.

Suppose you perform $k$ operations.

$$E_k\ldots E_1A=R$$

Then we have

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

To get the elmentary matrix, perform the same operation on the identity matrix.

Suppose the first operation is multiply the first row by $\frac12$.

$$E_1=\begin{bmatrix} \frac12 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$

$E_1^{-1}$ can be obtained by thinking about what is the reverse operation? It should be multiply the first row by $2$.

$$E_1^{-1}=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$$