###### Exercise 5

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 \(A^{-1}\text{.}\)

- \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*}
- \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*}
- \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*}