Question

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

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

Step-by-step solution

Reduce the given determinant

Interchange row 1 and row 2 to obtain:

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

Use the given statement

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

Conclusion

Therefore,

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