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 $-1$:
$$\left[\begin{array}{c}
R_2\\
R_1\\
R_3\\
\vdots\\
R_n
\end{array}\right].$$
