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


Linearity

This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.

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