Q46E Determine whether the statement ... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q46E (page 264)

Determine whether the statement 鈥淚f A is any symmetric $2 脳 2$ matrix, then there must exist a real number x such that the matrix $A - x I_{2}$ fails to be invertible.

Short Answer

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The solution is true.

Step by step solution

Explanation of the solution

Consider an orthogonal basis for the subspace V, say $\{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}\}$ and ${\overset{鈫�}{v}}_{3}$ is a vector perpendicular to the plane V.

Since A represents the orthogonal projection onto the plane V as follows.

role="math" localid="1659623630123" $A [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� 0] A S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� 0] 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� w h e r e 鈥� 鈥� S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] A S = [{\overset{鈫�}{v}}_{1} 鈥� {\overset{鈫�}{v}}_{2} 鈥� {\overset{鈫�}{v}}_{3}] [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] A S = S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Simplify further as follows.

role="math" localid="1659624009075" $A S = S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] S^{- 1} A S = S^{- 1} S [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Since S is orthogonal is $S^{T} = S^{- 1}$ .

$S^{T} A S = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

This implies that $S^{T} A S$ is a diagonal matrix.

Thus, the given statement is true.

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91影视

Determine whether the statement 鈥淚f A is any symmetric $2 脳 2$ matrix, then there must exist a real number x such that the matrix $A - x I_{2}$ fails to be invertible.

Short Answer

Step by step solution

Explanation of the solution

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91影视

Determine whether the statement 鈥淚f A is any symmetric 2脳2matrix, then there must exist a real number x such that the matrix A-xI2fails to be invertible.

Short Answer

Step by step solution

Explanation of the solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Geometry

Study anywhere. Anytime. Across all devices.

Determine whether the statement 鈥淚f A is any symmetric $2 脳 2$ matrix, then there must exist a real number x such that the matrix $A - x I_{2}$ fails to be invertible.