Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 7.3 : Augmented Matrices
3. For the following augmented matrix perform the indicated elementary row operations.
\[\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\0&7&{ - 2}&3\end{array}} \right]\]- \( - 9{R_{\,3}}\)
- \({R_{\,1}} \leftrightarrow {R_{\,2}}\)
- \({R_{\,3}} - {R_{\,1}} \to {R_{\,3}}\)
Show All Solutions Hide All Solutions
a Show SolutionThis operation is telling us to multiply all the entries in Row 3 of the augmented matrix by -9 so let’s do that.
\[\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\0&7&{ - 2}&3\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{ - 9{R_{\,3}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\0&{ - 63}&{18}&{ - 27}\end{array}} \right]\]b Show Solution
This operation is telling us to interchange Row 1 and Row 2 of the augmented matrix. Here is that work.
\[\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\0&7&{ - 2}&3\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,1}} \leftrightarrow {R_{\,2}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}4&0&7&{ - 1}\\{10}&{ - 1}&{ - 5}&1\\0&7&{ - 2}&3\end{array}} \right]\]c Show Solution
For this operation we are going to replace Row 3 with the results of taking the original entries from Row 3 and subtract from them the entries in Row 1.
\[\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\0&7&{ - 2}&3\end{array}} \right]\,\,\,\,\,\,\,\,\,\,\begin{array}{*{20}{c}}{{R_{\,3}} - {R_{\,1}} \to {R_{\,3}}}\\ \to \end{array}\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{rrr|r}{10}&{ - 1}&{ - 5}&1\\4&0&7&{ - 1}\\{ - 10}&8&3&2\end{array}} \right]\]Here are the individual computations for this operation.
\[\begin{align*}& {\mbox{Column 1 }}:\,\,\,0 - \left( {10} \right) = - 10\\ & {\mbox{Column 2 }}:\,\,\,7 - \left( { - 1} \right) = 8\\ & {\mbox{Column 3 }}:\,\,\, - 2 - \left( { - 5} \right) = 3\\ & {\mbox{Column 4 }}:\,\,\,3 - \left( 1 \right) = 2\end{align*}\]