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Q17E

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Found in: Page 165

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# 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|

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

See the step by step solution

## Step 1: 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|

## Step 2: 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.

## Step 3: Conclusion

Therefore,

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