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Q14E

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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

# Compute the determinants in Exercises 9-14 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.\left| {\begin{aligned}{*{20}{c}}{\bf{6}}&{\bf{3}}&{\bf{2}}&{\bf{4}}&{\bf{0}}\\{\bf{9}}&{\bf{0}}&{ - {\bf{4}}}&{\bf{1}}&{\bf{0}}\\{\bf{8}}&{ - {\bf{5}}}&{\bf{6}}&{\bf{7}}&{\bf{1}}\\{\bf{2}}&{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{4}}&{\bf{2}}&{\bf{3}}&{\bf{2}}&{\bf{0}}\end{aligned}} \right|

The value of the determinant is 6.

See the step by step solution

## Step 1: Expand the determinant about the fifth column

The determinant can be expanded as shown below:

\left| {\begin{aligned}{*{20}{c}}6&3&2&4&0\\9&0&{ - 4}&1&0\\8&{ - 5}&6&7&1\\2&0&0&0&0\\4&2&3&2&0\end{aligned}} \right| = {\left( { - 1} \right)^{3 + 5}} \cdot 1\left| {\begin{aligned}{*{20}{c}}6&3&2&4\\9&0&{ - 4}&1\\2&0&0&0\\4&2&3&2\end{aligned}} \right|

## Step 2: Expand the determinant about the third row

The determinant can be expanded as shown below:

\begin{aligned}{c}{\left( { - 1} \right)^{3 + 5}} \cdot 1\left| {\begin{aligned}{*{20}{c}}6&3&2&4\\9&0&{ - 4}&1\\2&0&0&0\\4&2&3&2\end{aligned}} \right| = \left\{ {{{\left( { - 1} \right)}^{3 + 1}} \cdot 2\left| {\begin{aligned}{*{20}{c}}3&2&4\\0&{ - 4}&1\\2&3&2\end{aligned}} \right|} \right\}\\ = 2\left| {\begin{aligned}{*{20}{c}}3&2&4\\0&{ - 4}&1\\2&3&2\end{aligned}} \right|\end{aligned}

## Step 3: Expand the determinant about the first column

The determinant can be expanded as shown below:

\begin{aligned}{c}2\left| {\begin{aligned}{*{20}{c}}3&2&4\\0&{ - 4}&1\\2&3&2\end{aligned}} \right| = 2\left\{ {{{\left( { - 1} \right)}^{1 + 1}} \cdot 3\left| {\begin{aligned}{*{20}{c}}{ - 4}&1\\3&2\end{aligned}} \right| + {{\left( { - 1} \right)}^{3 + 1}} \cdot 2\left| {\begin{aligned}{*{20}{c}}2&4\\{ - 4}&1\end{aligned}} \right|} \right\}\\ = 2\left\{ {3\left( { - 11} \right) + 2\left( {18} \right)} \right\}\\ = 6\end{aligned}

So, the value of the determinant is 6.