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Chapter 3: Q18E (page 165)

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

Short Answer

Expert verified

Hence, \(\left| {\begin{aligned}{*{20}{c}}d&e&f\\a&b&c\\g&h&i\end{aligned}} \right| = - 7\).

Step by step solution

01

Reduce the given determinant

Interchange row 1 and row 2 to obtain:

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

02

Use the given statement

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

03

Conclusion

Therefore,

\(\left| {\begin{aligned}{*{20}{c}}d&e&f\\a&b&c\\g&h&i\end{aligned}} \right| = - 7\)

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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.

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

Question: In Exercises 31鈥�36, mention an appropriate theorem in your explanation.

33. Let A and B be square matrices. Show that even thoughABand BAmay not be equal, it is always true that\(det{\rm{ }}AB = det{\rm{ }}BA\).

In Exercise 33-36, verify that \(\det EA = \left( {\det E} \right)\left( {\det A} \right)\)where E is the elementary matrix shown and \(A = \left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]\).

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

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.

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

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.

\[\left| {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{7}}&{ - {\bf{2}}}&{\bf{0}}&{\bf{0}}\\{\bf{2}}&{\bf{6}}&{\bf{3}}&{\bf{0}}\\{\bf{3}}&{ - {\bf{8}}}&{\bf{4}}&{ - {\bf{3}}}\end{array}} \right|\]
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