**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].$$