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Chapter 3: Q23E (page 165)

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

Expert verified

The determinant changes sign when the rows are swapped.

Step by step solution

01

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:

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

02

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.

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Most popular questions from this chapter

Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.

5. \(\begin{array}{c}{x_1} + {x_2} = 3\\ - 3{x_1} + 2{x_3} = 0\\{x_2} - 2{x_3} = 2\end{array}\)

Find the determinant in Exercise 18, where \(\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\).

18. \(\left| {\begin{aligned}{*{20}{c}}{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right|\)

In Exercises 27 and 28, A and B are \[n \times n\] matrices. Mark each statement True or False. Justify each answer.

27. a. A row replacement operation does not affect the determinant of a matrix.

b. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by \({\left( { - {\bf{1}}} \right)^r}\), where r is the number of row interchanges made during row reduction from A to U.

c. If the columns of A are linearly dependent, then \(det\left( A \right) = 0\).

d. \(det\left( {A + B} \right) = det{\rm{ }}A + det{\rm{ }}B\).

Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.

2. \(\begin{array}{l}4{x_1} + {x_2} = 6\\3{x_1} + 2{x_2} = 7\end{array}\)

Question: In Exercise 12, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

12. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{1}}&{\bf{3}}\\{ - {\bf{2}}}&{\bf{2}}&{\bf{1}}\\{\bf{0}}&{\bf{1}}&{\bf{1}}\end{array}} \right)\)
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