Question

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.

\(\left[ {\begin{array}{*{20}{c}}a&b&c\\{\bf{3}}&{\bf{2}}&{\bf{1}}\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}\\a&b&c\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right]\)

Short answer

The determinant changes sign when the rows are swapped.

Step-by-step solution

Find the determinant of the first matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}a&b&c\\3&2&1\\4&5&6\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}a&b&c\\3&2&1\\4&5&6\end{array}} \right| = a\left| {\begin{array}{*{20}{c}}2&1\\5&6\end{array}} \right| - b\left| {\begin{array}{*{20}{c}}3&1\\4&6\end{array}} \right| + c\left| {\begin{array}{*{20}{c}}3&2\\4&5\end{array}} \right|\\ = a\left( {12 - 5} \right) - b\left( {18 - 4} \right) + c\left( {15 - 8} \right)\\ = 7a - 14b + 7c\end{array}\)

Find the determinant of the second matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}3&2&1\\a&b&c\\4&5&6\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}3&2&1\\a&b&c\\4&5&6\end{array}} \right| = 3\left| {\begin{array}{*{20}{c}}b&c\\5&6\end{array}} \right| - 2\left| {\begin{array}{*{20}{c}}a&c\\4&6\end{array}} \right| + 1\left| {\begin{array}{*{20}{c}}a&b\\4&5\end{array}} \right|\\ = 3\left( {6b - 5c} \right) - 2\left( {6a - 4c} \right) + \left( {5a - 4b} \right)\\ = 18b - 15c - 12a + 8c + 5a - 4b\\ = - 7a + 14b - 7c\end{array}\)

So, the determinant changes signs when the rows are interchanged.