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

The given statement is true.

See the step by step solution

## Step 1: Consider the parameters

Consider the matrix.

$A=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

## Step 2: Compute the matrices

The matrices are,

${A}^{2}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\phantom{\rule{0ex}{0ex}}{A}^{3}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\phantom{\rule{0ex}{0ex}}{A}^{4}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

The series keeps on going.

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

Hence, the statement is true.