Difference between revisions of "031 Review Part 1, Problem 2"

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(Created page with "<span class="exam">True or false: If all the entries of a  <math style="vertical-align: 0px">7\times 7</math>  matrix  <math style="vertical-align: 0px">A</math...")
 
 
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<span class="exam">True or false: If all the entries of a &nbsp;<math style="vertical-align: 0px">7\times 7</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">7,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\text{det }A</math>&nbsp; must be &nbsp;<math style="vertical-align: 0px">7^7.</math>
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<span class="exam"> True or false: If a matrix &nbsp;<math style="vertical-align: 0px">A^2</math>&nbsp; is diagonalizable, then the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; must be diagonalizable as well.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;  
 
!Solution: &nbsp;  
 
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|-
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|First, we switch to the limit to <math style="vertical-align: 0px">x</math> so that we can use L'Hopital's rule.
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|Let &nbsp;<math style="vertical-align: -20px">A=   
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    \begin{bmatrix}
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          0 & 1  \\
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          0 & 0
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        \end{bmatrix}.</math>&nbsp;
 
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|-
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|So, we have
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|First, notice that
 
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|-
 
|
 
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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::<math style="vertical-align: -20px">A^2=   
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\displaystyle{\lim_{x \rightarrow \infty}\frac{3-2x^2}{5x^2 + x +1}} & \overset{L'H}{=} & \displaystyle{\lim_{x \rightarrow \infty}\frac{-4x}{10x+1}}\\
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    \begin{bmatrix}
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&&\\
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          0 & 0  \\
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& \overset{L'H}{=} & \displaystyle{\frac{-4}{10}}\\
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          0 & 0
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&&\\
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        \end{bmatrix},</math>&nbsp;
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& = & \displaystyle{-\frac{2}{5}}.
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|-
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\end{array}</math>
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|which is diagonalizable.
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|-
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|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a triangular matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are the entries on the diagonal.
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|-
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|Therefore, the only eigenvalue of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is &nbsp;<math style="vertical-align: -1px">0.</math>&nbsp; Additionally, there is only one linearly independent eigenvector.
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|-
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|Hence, &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is not diagonalizable and the statement is false.
 
|}
 
|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; &nbsp; &nbsp; False
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|&nbsp;&nbsp; &nbsp; &nbsp; FALSE
 
|}
 
|}
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[[031_Review_Part_1|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_1|'''<u>Return to Review Problems</u>''']]

Latest revision as of 11:17, 15 October 2017

True or false: If a matrix    is diagonalizable, then the matrix    must be diagonalizable as well.

Solution:  
Let   
First, notice that
 
which is diagonalizable.
Since    is a triangular matrix, the eigenvalues of    are the entries on the diagonal.
Therefore, the only eigenvalue of    is    Additionally, there is only one linearly independent eigenvector.
Hence,    is not diagonalizable and the statement is false.


Final Answer:  
       FALSE

Return to Review Problems

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