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Chapter 3: Q13E (page 165)

Compute the determinants in Exercises 9-14 by cofactor expnasions. At each step, choose a row or column that involves the least amount of computation.

\(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{ - {\bf{7}}}&{\bf{3}}&{ - {\bf{5}}}\\{\bf{0}}&{\bf{0}}&{\bf{2}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{\bf{3}}&{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{5}}&{\bf{0}}&{\bf{5}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{0}}&{\bf{9}}&{ - {\bf{1}}}&{\bf{2}}\end{array}} \right|\)

Short Answer

Expert verified

The value of the determinant is 6.

Step by step solution

01

Expand the determinant about the second row

The determinant can be expanded as shown below:

\(\left| {\begin{array}{*{20}{c}}4&0&{ - 7}&3&{ - 5}\\0&0&2&0&0\\7&3&{ - 6}&4&{ - 8}\\5&0&5&2&{ - 3}\\0&0&9&{ - 1}&2\end{array}} \right| = {\left( { - 1} \right)^{2 + 3}} \cdot 2\left| {\begin{array}{*{20}{c}}4&0&3&{ - 5}\\7&3&4&{ - 8}\\5&0&2&{ - 3}\\0&0&{ - 1}&2\end{array}} \right|\)

02

Expand the determinant about the second column

The determinant can be expanded as shown below:

\(\begin{array}{c}{\left( { - 1} \right)^{2 + 3}} \cdot 2\left| {\begin{array}{*{20}{c}}4&0&3&{ - 5}\\7&3&4&{ - 8}\\5&0&2&{ - 3}\\0&0&{ - 1}&2\end{array}} \right| = \left( { - 2} \right) \cdot \left\{ {{{\left( { - 1} \right)}^{2 + 2}} \cdot 3\left| {\begin{array}{*{20}{c}}4&3&{ - 5}\\5&2&{ - 3}\\0&{ - 1}&2\end{array}} \right|} \right\}\\ = \left( { - 6} \right)\left| {\begin{array}{*{20}{c}}4&3&{ - 5}\\5&2&{ - 3}\\0&{ - 1}&2\end{array}} \right|\end{array}\)

03

Expand the determinant about the first column

The determinant can be expanded as shown below:

\(\begin{array}{c}\left( { - 6} \right)\left| {\begin{array}{*{20}{c}}4&3&{ - 5}\\5&2&{ - 3}\\0&{ - 1}&2\end{array}} \right| = \left( { - 6} \right)\left\{ {{{\left( { - 1} \right)}^{1 + 1}} \cdot 4\left| {\begin{array}{*{20}{c}}2&{ - 3}\\{ - 1}&2\end{array}} \right| + {{\left( { - 1} \right)}^{2 + 1}} \cdot 5\left| {\begin{array}{*{20}{c}}3&{ - 5}\\{ - 1}&2\end{array}} \right|} \right\}\\ = \left( { - 6} \right)\left\{ {4 - 5} \right\}\\ = 6\end{array}\)

So, the value of the determinant is 6.

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Most popular questions from this chapter

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.

\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}a&b\\{kc}&{kd}\end{array}} \right]\)

Compute the determinants of the elementary matrices given in Exercise 25-30.

26. \(\left[ {\begin{aligned}{*{20}{c}}0&0&1\\0&1&0\\1&0&0\end{aligned}} \right]\).

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.

\[\left[ {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{1}}\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}k&{\bf{0}}&k\\{ - {\bf{3}}}&{\bf{4}}&{ - {\bf{4}}}\\{\bf{2}}&{ - {\bf{3}}}&{\bf{1}}\end{array}} \right]\]

Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9. How is \(det {A^{ - 1}}\) related to \(det A\)? Experiment with random \({\bf{n}} \times {\bf{n}}\) integer matrices for \(n = 4\), 5, and 6, and make a conjecture. Note:In the unlikely event that you encounter a matrix with a zero determinant, reduce

it to echelon form and discuss what you find.

Question: In Exercise 10, determine the values of the parameter s for which the system has a unique solution, and describe the solution.

10.

\(\begin{array}{c}s{x_{\bf{1}}} - {\bf{2}}{x_{\bf{2}}} = {\bf{1}}\\4s{x_{\bf{1}}} + {\bf{4}}s{x_{\bf{2}}} = {\bf{2}}\end{array}\)
See all solutions

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