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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 33-36, verify that $$\det EA = \left( {\det E} \right)\left( {\det A} \right)$$where E is the elementary matrix shown and $$A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$$.35. $$\left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]$$

It is verified that $$\det EA = \left( {\det E} \right)\left( {\det A} \right)$$.

See the step by step solution

## Step 1: Determine matrix $$EA$$

It is given that $$A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],{\rm{ }}E = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]$$.

Compute matrix $$EA$$ as shown below:

$\begin{array}{c}EA = \left[ {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{0 + c}&{0 + d}\\{a + 0}&{b + 0}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}c&d\\a&b\end{array}} \right]\end{array}$

## Step 2: Verify that $$\det EA = \left( {\det E} \right)\left( {\det A} \right)$$

The determinants of matrices E and A are shown below:

$\begin{array}{c}\det E = \left| {\begin{array}{*{20}{c}}0&1\\1&0\end{array}} \right|\\ = 0 - 1\\ = - 1\\\det A = \left| {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right|\\ = ad - bc\end{array}$

The determinant of matrix $$EA$$ is shown below:

$$\begin{array}{c}\det EA = \left| {\begin{array}{*{20}{c}}c&d\\a&b\end{array}} \right|\\ = cb - da\\ = - 1\left( {ad - bc} \right)\\ = \left( {\det E} \right)\left( {\det A} \right)\end{array}$$

Thus, it is verified that $$\det EA = \left( {\det E} \right)\left( {\det A} \right)$$.