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Chapter 3: Q20E (page 165)

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

20. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{{\bf{d}} + {\bf{3g}}}&{{\bf{e}} + {\bf{3h}}}&{{\bf{f}} + {\bf{3i}}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right|\]

Short Answer

Expert verified

Hence \[\left| {\begin{array}{*{20}{c}}a&b&c\\{d + 3g}&{e + 3h}&{f + 3i}\\g&h&i\end{array}} \right| = 7\].

Step by step solution

01

Reduce the given determinant

At row 2, add \[ - 3\] times row 3 to row 2 to obtain:

\[\left| {\begin{array}{*{20}{c}}a&b&c\\{d + 3g}&{e + 3h}&{f + 3i}\\g&h&i\end{array}} \right| = \left| {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right|\]

02

Use the given statement

The given statement is \[\left| {\begin{array}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{array}} \right| = 7\].

03

Conclusion

Therefore,

\[\left| {\begin{array}{*{20}{c}}a&b&c\\{d + 3g}&{e + 3h}&{f + 3i}\\g&h&i\end{array}} \right| = 7\]

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Most popular questions from this chapter

Find the determinant in Exercise 17, where \[\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\].

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

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.

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

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

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{1}}}&{\bf{0}}&{ - {\bf{2}}}\\{\bf{0}}&{\bf{2}}&{ - {\bf{4}}}&{ - {\bf{2}}}&{ - {\bf{6}}}\\{ - {\bf{2}}}&{ - {\bf{6}}}&{\bf{2}}&{\bf{3}}&{{\bf{10}}}\\{\bf{1}}&{\bf{5}}&{ - {\bf{6}}}&{\bf{2}}&{ - {\bf{3}}}\\{\bf{0}}&{\bf{2}}&{ - {\bf{4}}}&{\bf{5}}&{\bf{9}}\end{array}} \right|\)

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.

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