Short Answer

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 by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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.

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra and its Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Expand the determinant about the fifth column

Step 2: Expand the determinant about the third row

Step 3: Expand the determinant about the first column

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra and its Applications

Answers without the blur.

Short Answer

Step by Step Solution

Step 1: Expand the determinant about the fifth column

Step 2: Expand the determinant about the third row

Step 3: Expand the determinant about the first column

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths