###### Exercise5

If a matrix $$A$$ is invertible, there is a sequence of row operations that transform $$A$$ into the identity matrix $$I\text{.}$$ We have seen that every row operation can be performed by matrix multiplication. If the $$j^{th}$$ step in the Gaussian elimination process is performed by multiplying by $$E_j\text{,}$$ then we have

\begin{equation*} E_p\ldots E_2E_1 A = I\text{,} \end{equation*}

which means that

\begin{equation*} A^{-1} = E_p\ldots E_2E_1\text{.} \end{equation*}

For each of the following matrices, find a sequence of row operations that transforms the matrix to the identity $$I\text{.}$$ Write the matrices $$E_j$$ that perform the steps and use them to find $$A^{-1}\text{.}$$

1. \begin{equation*} A = \left[\begin{array}{rrr} 0 \amp 2 \amp 0 \\ -3 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}
2. \begin{equation*} A = \left[\begin{array}{rrrr} 1 \amp 0 \amp 0 \amp 0 \\ 2 \amp 1 \amp 0 \amp 0 \\ 0 \amp -3 \amp 1 \amp 0 \\ 0 \amp 0 \amp 2 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}
3. \begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 1 \amp 1 \\ 0 \amp 1 \amp 1 \\ 0 \amp 0 \amp 2 \\ \end{array}\right]\text{.} \end{equation*}
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