Exercise 4.6.1

For each of the following pairs of matrices, suppose a row operation is applied to $A$ and the result is $B$. Find the elementary matrix $E$ that represents this row operation.

1. $$A=\left[\begin{array}{rr}2&3\\ 1&2\end{array}\right],\quad B=\left[\begin{array}{rr}1&2\\ 2&3\end{array}\right].$$
2. $$A=\left[\begin{array}{rr}4&0\\ 2&1\end{array}\right],\quad B=\left[\begin{array}{rr}8&0\\ 2&1\end{array}\right].$$
3. $$A=\left[\begin{array}{rr}1&-3\\ 0&5\end{array}\right],\quad B=\left[\begin{array}{rr}1&-3\\ 2&-1\end{array}\right].$$