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## Interchange of two rows can be accomplished by elementary row operations of other types

Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.7

Solution: Write the matrix as
$$\left[\begin{array}{c} R_1\\ R_2\\ R_3\\ \vdots\\ R_n \end{array}\right].$$WOLOG we'll show how to exchange rows $R_1$ and $R_2$. First add $R_2$ to $R_1$:
$$\left[\begin{array}{c} R_1+R_2\\ R_2\\ R_3\\ \vdots\\ R_n \end{array}\right].$$Next subtract row one from row two:
$$\left[\begin{array}{c} R_1+R_2\\ -R_1\\ R_3\\ \vdots\\ R_n \end{array}\right].$$Next add row two to row one again
$$\left[\begin{array}{c} R_2\\ -R_1\\ R_3\\ \vdots\\ R_n \end{array}\right].$$Finally multiply row two by $-1$:
$$\left[\begin{array}{c} R_2\\ R_1\\ R_3\\ \vdots\\ R_n \end{array}\right].$$