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Expert-verified Found in: Page 1 ### 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 Exercises 31, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.31. $$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]$$, $$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]$$

The elementary row operation that transforms the first matrix into the second matrix is performed by replacing $${R_3}$$ by $${R_3} + \left( { - 4} \right){R_1}$$.

The elementary row operation that transforms the second matrix into the first matrix is performed by replacing $${R_3}$$ by $${R_3} + 4{R_1}$$.

See the step by step solution

## Step 1: Apply row operation

A basic principle of this section is that row operations do not affect the solution set of a linear system.

Use the $${x_1}$$ term in the first equation to eliminate the $$4{x_1}$$ term from the third equation. Add $$- 4$$ times row 1 to row 3.

$$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]$$

## Step 2: Write the elementary row operation

From the obtained augmented matrix $$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]$$, it is transformed into the second matrix.

The elementary row operation used to convert the first matrix into the second matrix is shown below:

Replace $${R_3}$$ by $${R_3} + \left( { - 4} \right){R_1}$$. Here, $$R$$ represents a row.

## Step 3: Apply row operation

A basic principle of this section is that row operations do not affect the solution set of a linear system.

Use the $${x_1}$$ term in the first equation to add the $$4{x_1}$$ term in the third equation. Add $$4$$ times row 1 to row 3.

$$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]$$

## Step 4: Write the elementary row operation

From the obtained augmented matrix $$\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]$$, it is transformed into the first matrix.

The elementary row operation used to convert the second matrix into the first matrix is shown below:

Replace $${R_3}$$ by $${R_3} + 4{R_1}$$. Here, $$R$$ represents a row.

Thus, the elementary row operation that transforms the first matrix into the second matrix is performed by replacing $${R_3}$$ by $${R_3} + \left( { - 4} \right){R_1}$$. For the reverse, replace $${R_3}$$ by $${R_3} + 4{R_1}$$. ### Want to see more solutions like these? 