StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q24E

Expert-verifiedFound in: Page 165

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

**In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.**

** **

\[\left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{1}}\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}k&{\bf{0}}&k\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right]\]

The row operation scales row 1 by *k* and multiplies the determinant by *k*.

The **determinant** of the matrix \(\left[ {\begin{array}{*{20}{c}}1&0&1\\{ - 3}&4&{ - 4}\\2&{ - 3}&1\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}1&0&1\\{ - 3}&4&{ - 4}\\2&{ - 3}&1\end{array}} \right| = 1\left| {\begin{array}{*{20}{c}}4&{ - 4}\\{ - 3}&1\end{array}} \right| - 0 + 1\left| {\begin{array}{*{20}{c}}{ - 3}&4\\2&{ - 3}\end{array}} \right|\\ = \left( {4 - 12} \right) + \left( {9 - 8} \right)\\ = - 7\end{array}\)

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}k&0&k\\{ - 3}&4&{ - 4}\\2&{ - 3}&1\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}k&0&k\\{ - 3}&4&{ - 4}\\2&{ - 3}&1\end{array}} \right| = k\left| {\begin{array}{*{20}{c}}4&{ - 4}\\{ - 3}&1\end{array}} \right| - 0 + k\left| {\begin{array}{*{20}{c}}{ - 3}&4\\2&{ - 3}\end{array}} \right|\\ = k\left( {4 - 12} \right) + k\left( {9 - 8} \right)\\ = - 7k\end{array}\)

So, the row operation scales row 1 by *k* and multiplies the determinant by *k*.

94% of StudySmarter users get better grades.

Sign up for free