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Q31E

Expert-verifiedFound in: Page 1

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 \({R_3} + 4{R_1}\).

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.

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

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.

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.

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

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 \({R_3} + 4{R_1}\).

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