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Chapter 3: Q17E (page 165)

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

Short Answer

Expert verified

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

Step by step solution

01

Reduce the given determinant

At row 1, add \[ - 1\] time row 2 to row 1 to obtain:

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

02

Use the given statement

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

03

Conclusion

Therefore,

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

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Most popular questions from this chapter

Compute \(det{\rm{ }}{B^4}\), where \(B = \left[ {\begin{aligned}{{}{}}1&0&1\\1&1&2\\1&2&1\end{aligned}} \right]\).

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

36. Let U be a square matrix such that \({U^T}U = I\). Show that\(det{\rm{ }}U = \pm 1\).

In Exercises 31-36, mention an appropriate theorem in your explanation.

36. Find a formula for \(\det \left( {rA} \right)\) when \(A\) is an\(n \times n\) matrix.

Compute the determinant in Exercise 5 using a cofactor expansion across the first row.

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

Compute the determinants of the elementary matrices given in Exercise 25-30.

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