Write the matrix \(A\) that corresponds to the system above.
Swap rows \(R_1\) and \(R_2\) and write down the new system of equations and the elementary matrix \(E_1=E_{12}\) that performs the swap.
Continue to perform row operations on the system until it is in row-echelon form. At each step write down and keep track of the corresponding elementary matrix; that is, \(E_1=E_{12}\) (elementary matrix corresponding to your first row operation), \(E_2\text{,...,}\)\(E_m\) (elementary matrix corresponding to the last row operation). Note each group member should do this on their own (it’s ok if you do it in different order - just check each person all verify in part (e)).
Multiply the elementary matrices together in reverse order \(E=E_m\cdots E_2E_1.\)
Perform the multiplication \(EA\) and verify it corresponds to the row-echelon form you found in part Item 2.c.
Let \(E\in \R^{3 \times 3}\) be the matrix that swaps rows \(1\) and \(3\text{.}\) Write down the matrix \(E_{13}\) and show that \(E_{13}^{-1}=E_{13}\text{.}\)
Let \(E \in \R^{3 \times 3}\) be the matrix that performs a scalar multiplication of row \(R_2\) by \(5\text{,}\)\(E_{(5)22}\text{.}\) Write down \(E_{(5)22}\) and make a guess as to its inverse. Check your answer and keep guessing until you find the inverse. Do you notice a relationship between \(E_{(5)22}\) and \(E_{(5)22}^{-1}\text{?}\)
Let \(E\in \R^{3 \times 3}\) be the matrix that performs the operation \(E_{(2)13}\text{.}\) Write down the matrix \(E_{(2)13}\) and find its inverse using a guess and check method. Do you notice a relationship between \(E_{(2)13}\) and \(E_{(2)13}^{-1}\text{?}\)
