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Q6E

Expert-verified

Linear Algebra and its Applications

Found in: Page 165

Book edition 5th

Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald

Pages 483 pages

ISBN 978-03219822384

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Short Answer

Compute the determinant in Exercise 6 using a cofactor expansion across the first row.
**6. \(\left| {\begin{aligned}{*{20}{c}}{\bf{5}}&{ - {\bf{2}}}&{\bf{2}}\\{\bf{0}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{2}}&{ - {\bf{4}}}&{\bf{7}}\end{aligned}} \right|\)**

Thus, \(\left| {\begin{aligned}{*{20}{c}}5&{ - 2}&2\\0&3&{ - 3}\\2&{ - 4}&7\end{aligned}} \right| = 45\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the determinant formula

The determinant computed by a cofactor expansion across the i^th row is

\(\det A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + \cdots + {a_{in}}{C_{in}}\).

Here, A is an \(n \times n\) matrix, and \({C_{ij}} = {\left( { - 1} \right)^{i + j}}{A_{ij}}\).

Step 2: Use the cofactor expansion across the first row

\(\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}5&{ - 2}&2\\0&3&{ - 3}\\2&{ - 4}&7\end{aligned}} \right| = {a_{11}}{C_{11}} + {a_{12}}{C_{12}} + {a_{13}}{C_{13}}\\ = {a_{11}}{\left( { - 1} \right)^{1 + 1}}\det {A_{11}} + {a_{12}}{\left( { - 1} \right)^{1 + 2}}\det {A_{12}} + {a_{13}}{\left( { - 1} \right)^{1 + 3}}\det {A_{13}}\\ = 5\left| {\begin{aligned}{*{20}{c}}3&{ - 3}\\{ - 4}&7\end{aligned}} \right| - \left( { - 2} \right)\left| {\begin{aligned}{*{20}{c}}0&{ - 3}\\2&7\end{aligned}} \right| + 2\left| {\begin{aligned}{*{20}{c}}0&3\\2&{ - 4}\end{aligned}} \right|\\ = 5\left( 9 \right) + 2\left( 6 \right) + 2\left( { - 6} \right)\\ = 45 + 12 - 12\\ = 45\end{aligned}\)

Step 3: Conclusion

Hence, the given determinant using a cofactor expansion across the first row is 45.

Most popular questions for Math Textbooks

Question: Use Cramer’s rule to compute the solutions of the systems in Exercises1-6.

2. \(\begin{array}{l}4{x_1} + {x_2} = 6\\3{x_1} + 2{x_2} = 7\end{array}\)

In Exercises 21–23, use determinants to find out if the matrix is invertible.

21. \(\left( {\begin{aligned}{*{20}{c}}2&6&0\\1&3&2\\3&9&2\end{aligned}} \right)\)

In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

\(\left[ {\begin{array}{*{20}{c}}a&b&c\\{\bf{3}}&{\bf{2}}&{\bf{1}}\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}&{\bf{1}}\\a&b&c\\{\bf{4}}&{\bf{5}}&{\bf{6}}\end{array}} \right]\)

Compute the determinant in Exercise 7 using a cofactor expansion across the first row.

7. \[\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{3}}&{\bf{0}}\\{\bf{6}}&{\bf{5}}&{\bf{2}}\\{\bf{9}}&{\bf{7}}&{\bf{3}}\end{array}} \right|\]

Question: In Exercise 14, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

14. \(\left( {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{1}}}&{\bf{2}}\\{\bf{0}}&{\bf{2}}&{\bf{1}}\\{\bf{2}}&{\bf{0}}&{\bf{4}}\end{array}} \right)\)

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