031 Review Part 3, Problem 8

Give an example of a  $3\times 3$  matrix  $A$  with eigenvalues 5,-1 and 3.

Foundations:
The eigenvalues of a diagonal matrix are the entries on the diagonal.

Solution:

One example of such a matrix is
$A=\left[{\begin{array}{ccc}5&0&0\\0&-1&0\\0&0&3\end{array}}\right].$
Since  $A$  is a diagonal matrix, the eigenvalues of  $A$  are the entries on the diagonal.
Hence, the eigenvalues of  $A$  are  $5,-1,3.$

Final Answer:
One example is  $A=\left[{\begin{array}{ccc}5&0&0\\0&-1&0\\0&0&3\end{array}}\right].$

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