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Q62E

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Found in: Page 109

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# TRUE OR FALSE?For every transition matrix A there exists a nonzero vector $\stackrel{\mathbf{\to }}{\mathbf{x}}$ such that ${\mathbit{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{=}}\stackrel{\mathbf{\to }}{\mathbf{x}}$.

The statement is false.

See the step by step solution

## Step 1: Consider the parameters.

Consider the system.

$Ax=x$

## Step 2: Consider the condition.

The system is $\begin{array}{l}Ax=x\end{array}$implies,

$\begin{array}{l}Ax=x\\ ⇒Ax-x=0\\ ⇒\left(A-l\right)x=0\end{array}$

Let any matrix $B=A-l$ then it is written as,

$⇒Bx=0$

This implies, $Bx=0\equiv Ax=x$ .

Thus, the system will have infinitely many solutions.

Hence, the statement is false.