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Q61E

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Found in: Page 109

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

TRUE OR FALSE?
There is a transition matrix A such that $\underset{m \to \infty}{l i m} A^{m}$ fails to exist.

The given statement is true.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Consider the parameters

Consider the matrix.

$A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$

Step 2: Compute the matrices

The matrices are,

$A^{2} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) A^{3} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) A^{4} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$

The series keeps on going.

But, $A^{m - \infty}$ does not exist.

Hence, the statement is true.

Most popular questions for Math Textbooks

Let $A = [\begin{matrix} 1 & 10 \\ - 3 & 12 \end{matrix}]$ in all parts of this problem.

(a) Find the scalar $λ$ such that the matrix $A - {λl}_{2}$ fails to be invertible. There are two solutions; choose one and use it in parts (b) and (c).

(b) For the $λ$ you choose in part (a), find a non-zero vector $\overset{⇀}{x}$ such that

role="math" localid="1659697491583" $(A - {λl}_{2}) \overset{⇀}{x} = \overset{⇀}{0}$

(c) Note that the equation $(A - {λl}_{2}) \overset{⇀}{x} = \overset{⇀}{0}$ can be written as.

$A - λ \overset{⇀}{x} = \overset{⇀}{0} o r A \overset{⇀}{x} = λ \overset{⇀}{x}$ Check that the equation $A \overset{⇀}{x} = λ \overset{⇀}{x}$ holds for your $λ$ from part (a) and your $\overset{⇀}{x}$ from part (b).

TRUE OR FALSE?

$A = [\begin{matrix} 1 & 1 / 2 \\ 0 & 1 / 2 \end{matrix}]$ is a regular transition matrix.

TRUE OR FALSE?

There exists an invertible matrix S such that $S^{- 1} (\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix})$ is a diagonal matrix.

If A is any invertible $n \times n$ matrix, then A commutes with $A^{- 1}$ .

If A and B are two 4 × 3 matrices such that $A \vec{v} = B \vec{v}$ for all vectors $\vec{v}$ in $ℝ^{3}$ , then matrices A and B must be equal.

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Short Answer

TRUE OR FALSE?
There is a transition matrix A such that $\underset{m \to \infty}{l i m} A^{m}$ fails to exist.

Step by Step Solution

Step 1: Consider the parameters

Step 2: Compute the matrices

Most popular questions for Math Textbooks

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE?There is a transition matrix A such that limm→∞Am fails to exist.

Step by Step Solution

Step 1: Consider the parameters

Step 2: Compute the matrices

Most popular questions for Math Textbooks

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TRUE OR FALSE?
There is a transition matrix A such that $\underset{m \to \infty}{l i m} A^{m}$ fails to exist.