Question

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

Short answer

The elementary row operation that transforms the first matrix into the second matrix is performedby 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 \({R_3} + 4{R_1}\).

Step-by-step solution

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.

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

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.

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.

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

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