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