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Q5E

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Linear Algebra and its Applications
Found in: Page 165
Linear Algebra and its Applications

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

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Short Answer

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{5}}&{ - {\bf{4}}}\\{ - {\bf{1}}}&{ - {\bf{4}}}&{\bf{5}}\\{ - {\bf{2}}}&{ - {\bf{8}}}&{\bf{7}}\end{array}} \right|\)

The value of the determinant is \( - 3\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Apply the row operation on the determinant

Apply the row operation to reduce the determinant into the echelon form.

At row 3, multiply row 1 by 2 and add it to row 3, i.e., \({R_3} \to {R_3} + 2{R_1}\).

\(\left| {\begin{array}{*{20}{c}}1&5&{ - 4}\\{ - 1}&{ - 4}&5\\0&2&{ - 1}\end{array}} \right|\)

Step 2: Apply the row operation on the determinant

At row 2, add rows 1 and 2, i.e., \({R_2} \to {R_2} + {R_1}\).

\(\left| {\begin{array}{*{20}{c}}1&5&{ - 4}\\0&1&1\\0&2&{ - 1}\end{array}} \right|\)

Step 3: Apply the row operation on the determinant

At row 3, multiply row 2 by 2 and subtract it from row 3.

\(\left| {\begin{array}{*{20}{c}}1&5&{ - 4}\\0&1&1\\0&0&{ - 3}\end{array}} \right|\)

Step 4: Find the value of the determinant

For a triangular matrix, the determinant is the product of diagonal elements.

\(\begin{array}{c}\det = \left( 1 \right)\left( 1 \right)\left( { - 3} \right)\\ = - 3\end{array}\)

So, the value of the determinant is \( - 3\).

Most popular questions for Math Textbooks

Question: 18. Suppose that all the entries in A are integers and \({\bf{det}}\,A = {\bf{1}}\). Explain why all the entries in \({A^{ - {\bf{1}}}}\) are integers.

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.

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

Question: 13. Show that if A is invertible, then adj A is invertible, and \({\left( {adj\,A} \right)^{ - {\bf{1}}}} = \frac{{\bf{1}}}{{detA}}A\).

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

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

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

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