Chapter 2.4, Problem 82E

Consider the matrix $E=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]$ and an arbitrary $3\times 3$ matrix $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right]$ .

a. Compute EA. Comment on the relationship between A and EA. in terms of the technique of elimination we learned in Section 1.2.
b. Consider the matrix $E=\left[\begin{array}{ccc}1& 0& 0\\ 0& \frac{1}{4}& 0\\ 0& 0& 1\end{array}\right]$ and an arbitrary $3\times 3$ matrix A. Compute EA. Comment on the relationship between A and EA.
c. Can you think of a $3\times 3$ matrix E such that EA is obtained from A by swapping the last two rows (for any $3\times 3$ matrix A)?
d. The matrices of the forms introduced in parts (a), (b), and (c) are called elementary: An $n\times n$ matrix E is elementary if it can be obtained from $I_n$ , by per forming one of the three elementary row operations on $I_n$ . Describe the format of the three types of elementary matrices.