Difference between revisions of "031 Review Part 3, Problem 8"

Revision as of 20:30, 10 October 2017

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 $A=\left[{\begin{array}{ccc}5&0&0\\0&-1&0\\0&0&3\end{array}}\right].$

@@ Line 5: / Line 5: @@
 !Foundations: &nbsp;
 |-
-|
+|The eigenvalues of a diagonal matrix are the entries on the diagonal.
 |}
@@ Line 12: / Line 12: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
+! &nbsp;
+|-
+|One example of such a matrix is
 |-
 |
-|}
+::<math>A=\left[\begin{array}{ccc}
+& 0 & 0\\
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+& -1 & 0\\
-!Step 2: &nbsp;
+& 0 & 3
+         \end{array}\right].</math>
+|-
+|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a diagonal matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are the entries on the diagonal.
 |-
-|
+|Hence, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">5,-1,3.</math>
 |}
@@ Line 27: / Line 32: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp; One example is &nbsp;<math style="vertical-align: -31px">A=\left[\begin{array}{ccc}
+& 0 & 0\\
+& -1 & 0\\
+& 0 & 3
+         \end{array}\right].</math>
 |}
 [[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]