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Expert-verified Found in: Page 165 ### Linear Algebra and its Applications

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{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right]

The row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant.

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## Step 1: Find the determinant of the first matrix

The determinant of the matrix \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right] can be calculated as shown below:

\left| {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right| = ad - bc

## Step 2: Find the determinant of the second matrix

The determinant of the matrix \left[ {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right] can be calculated as shown below:

\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right| = bc - ad\\ = - \left( {ad - bd} \right)\end{aligned}

So, the row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant. ### Want to see more solutions like these? 