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Chapter 3: Q4E (page 165)

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

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

Short Answer

Expert verified

The row operation does not change the value of the determinant.

Step by step solution

01

Recall the property of the determinant

When the row operation is applied to the determinant, its value does not change.

02

Apply the property for the given determinants

For the determinants, row operation is applied on row 3, i.e., \({R_2} \to {R_2} - 2{R_1}\).

So, the row operation does not change the value of the determinant.

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Most popular questions from this chapter

Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.

32. What is the determinant of an elementary scaling matrix with k on the diagonal?

Let \(u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\), and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\{\bf{0}}\end{aligned}} \right]\), where a, b, and c are positive integers (for simplicity). Compute the area of the parallelogram determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\), and \[\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\]. Draw a picture and explain what you find.

In Exercises 21–23, use determinants to find out if the matrix is invertible.

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

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

Find the determinant in Exercise 16, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

16. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{5d}}}&{{\bf{5e}}}&{{\bf{5f}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]
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