Exercise 5

If a matrix $A$ is invertible, there is a sequence of row operations that transforms $A$ into the identity matrix $I$. 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 $E_j$, then we have $$E_p\cdots E_2E_1 A = I$$, which means that $$A^{-1} = E_p\cdots E_2E_1$$. For each of the following matrices, find a sequence of row operations that transforms the matrix to the identity $I$. Write the matrices $E_j$ that perform the steps and use them to find $A^{-1}$.

1. $$A = \left[\begin{array}{rrr} 0 & 2 & 0 \\ -3 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$$.

2. $$A = \left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & -3 & 1 & 0 \\ 0 & 0 & 2 & 1 \\ \end{array}\right]$$.

3. $$A = \left[\begin{array}{rrr} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \\ \end{array}\right]$$.