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Q12SE

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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

Question: 12. Use the concept of area of a parallelogram to write a statement about a \(2 \times 2\) matrix A that is true if and only if A is invertible.

The area of a parallelogram is represented by a \(2 \times 2\) matrix A, i.e., its area is given by \(\left| {\det A} \right|\), provided A is invertible.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Use the fact of nonzero area

In this concept, the parallelogram contains four vectors, which are \(0,{v_1} \ne 0,{v_2} \ne 0,\) and \({v_3} \ne 0\), such that one of \({v_1},{v_2},\) and \({v_3}\) is the sum of the other two vectors if and only if the columns of A have nonzero area.

Step 2: Use the fact of invertible

The determinant of A is nonzero. It happens if and only if A is invertible.

Step 3: Conclusion

Hence, the area of a parallelogram is represented by a \(2 \times 2\) matrix A i.e., its area is given by \(\left| {\det A} \right|\), provided A is invertible.

Most popular questions for Math Textbooks

Question: In Exercise 7, determine the values of the parameter s for which the system has a unique solution, and describe the solution.

7.

\(\begin{array}{c}{\bf{6}}s{x_{\bf{1}}} + {\bf{4}}{x_{\bf{2}}} = {\bf{5}}\\{\bf{9}}{x_{\bf{1}}} + {\bf{2}}s{x_{\bf{2}}} = - {\bf{2}}\end{array}\)

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{4}}}\\{\bf{2}}&{\bf{0}}&{ - {\bf{3}}}\\{\bf{3}}&{ - {\bf{5}}}&{\bf{2}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{4}}}\\{\bf{0}}&{ - {\bf{6}}}&{\bf{5}}\\{\bf{3}}&{ - {\bf{5}}}&{\bf{2}}\end{array}} \right|\)

Question: In Exercise 15, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

15. \(\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{0}}&{\bf{0}}\\{ - {\bf{1}}}&{\bf{1}}&{\bf{0}}\\{ - {\bf{2}}}&{\bf{3}}&{ - {\bf{1}}}\end{array}} \right)\)

In Exercises 39 and 40, \(A\) is an \(n \times n\) matrix. Mark each statement True or False. Justify each answer.

39.

a. An \(n \times n\) determinant is defined by determinants of \(\left( {n - 1} \right) \times \left( {n - 1} \right)\) submatrices.

b. The \(\left( {i,j} \right)\)-cofactor of a matrix \(A\) is the matrix \({A_{ij}}\) obtained by deleting from A its \(i{\mathop{\rm th}\nolimits} \) row and \[j{\mathop{\rm th}\nolimits} \]column.

Find the determinant in Exercise 15, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

15. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{{\bf{3g}}}&{{\bf{3h}}}&{{\bf{3i}}}\end{array}} \right|\]
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