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### Linear Algebra With Applications

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

# For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.$\left[\begin{array}{ccc}-1& -1& -1\\ -1& -1& -1\\ -1& -1& -1\end{array}\right]$

Eigenvalues are:

${\lambda }_{1,2}=0,\mathrm{almu}\left(0\right)=2\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}_{3}=3,\mathrm{almu}\left(3\right)=3$

See the step by step solution

## Step 1: Eigenvalues

• In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by ${\mathbit{\lambda }}$, is the factor by which the eigenvector is scaled.
• Eigenvalues of a triangular matrix are its diagonal matrix.

## Step 2: Finding all real eigenvalues, with their algebraic multiplicities:

We can clearly see that,

$\mathrm{det}\left(\mathrm{A}-\mathrm{\lambda }\right)=0\phantom{\rule{0ex}{0ex}}\left|\begin{array}{ccc}-1-\mathrm{\lambda }& -1& -1\\ -1& -1-\mathrm{\lambda }& -1\\ -1& -1& -1-\mathrm{\lambda }\end{array}\right|=0\phantom{\rule{0ex}{0ex}}{\left(1-\mathrm{\lambda }\right)}^{3}-1-1-3\left(-1-\mathrm{\lambda }\right)=0\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}^{2}\left(\mathrm{\lambda }-3\right)=0\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}_{1,2}=0,{\mathrm{\lambda }}_{3}=3$

${\mathrm{\lambda }}_{1,2}=0,\mathrm{almu}\left(0\right)=2\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}_{3}=3,\mathrm{almu}\left(0\right)=3$