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Chapter 3: Q14E (page 165)

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

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

Short Answer

Expert verified

The value of the determinant is 6.

Step by step solution

01

Expand the determinant about the fifth column

The determinant can be expanded as shown below:

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

02

Expand the determinant about the third row

The determinant can be expanded as shown below:

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

03

Expand the determinant about the first column

The determinant can be expanded as shown below:

\(\begin{aligned}{c}2\left| {\begin{aligned}{*{20}{c}}3&2&4\\0&{ - 4}&1\\2&3&2\end{aligned}} \right| = 2\left\{ {{{\left( { - 1} \right)}^{1 + 1}} \cdot 3\left| {\begin{aligned}{*{20}{c}}{ - 4}&1\\3&2\end{aligned}} \right| + {{\left( { - 1} \right)}^{3 + 1}} \cdot 2\left| {\begin{aligned}{*{20}{c}}2&4\\{ - 4}&1\end{aligned}} \right|} \right\}\\ = 2\left\{ {3\left( { - 11} \right) + 2\left( {18} \right)} \right\}\\ = 6\end{aligned}\)

So, the value of the determinant is 6.

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Most popular questions from this chapter

The expansion of a \({\bf{3}} \times {\bf{3}}\) determinant can be remembered by the following device. Write a second type of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals.

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15-18. Warning: This trick does not generalize in any reasonable way to \({\bf{4}} \times {\bf{4}}\) or larger matrices.

15. \(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{0}}&{\bf{4}}\\{\bf{2}}&{\bf{3}}&{\bf{2}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{2}}}\end{array}} \right|\)

Compute the determinant in Exercise 1 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

  1. \(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{0}}&{\bf{4}}\\{\bf{2}}&{\bf{3}}&{\bf{2}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{1}}}\end{aligned}} \right|\)

Question: In Exercise 7, determine the values of the parameter s for which the system has a unique solution, and describe the solution.

7.

\(\begin{array}{c}{\bf{6}}s{x_{\bf{1}}} + {\bf{4}}{x_{\bf{2}}} = {\bf{5}}\\{\bf{9}}{x_{\bf{1}}} + {\bf{2}}s{x_{\bf{2}}} = - {\bf{2}}\end{array}\)

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

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

Question: In Exercise 19, find the area of the parallelogram whose vertices are listed.

19. \(\left( {0,0} \right),\left( {5,2} \right),\left( {6,4} \right),\left( {11,6} \right)\)
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