Difference between revisions of "031 Review Part 1, Problem 7"

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!Solution:    
 
!Solution:    
 
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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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|If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is not invertible, then &nbsp;<math style="vertical-align: 0px">\text{det } A=0.</math>
 
|-
 
|-
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|So, we have
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|Since &nbsp;<math style="vertical-align: -5px">C=AB,</math>&nbsp; we have
 
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|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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::<math>\begin{array}{rcl}
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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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\displaystyle{\text{det}C} & = & \displaystyle{\text{det} (AB)}\\
 
&&\\
 
&&\\
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& \overset{L'H}{=} & \displaystyle{\frac{-4}{10}}\\
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& = & \displaystyle{(\text{det} A) \cdot (\text{det} B)}\\
 
&&\\
 
&&\\
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& = & \displaystyle{-\frac{2}{5}}.
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& = & \displaystyle{0\cdot (\text{det} B)}\\
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\end{array}</math>
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&&\\
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& = & \displaystyle{0.}
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\end{array}</math>  
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|-
 +
|Since &nbsp;<math style="vertical-align: -5px">\text{det } C =0,</math>&nbsp; we know &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is not invertible, which is a contradiction.
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|-
 +
|So, &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; must be invertible and the statement is true.
 
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|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|&nbsp;&nbsp; &nbsp; &nbsp; False
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|&nbsp;&nbsp; &nbsp; &nbsp; TRUE
 
|}
 
|}
 
[[031_Review_Part_1|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 13:33, 9 October 2017

True or false: Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=AB}   for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4}   matrices  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B.}   If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}   is invertible, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is invertible.

Solution:  
If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is not invertible, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det } A=0.}
Since  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=AB,}   we have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\text{det}C} & = & \displaystyle{\text{det} (AB)}\\ &&\\ & = & \displaystyle{(\text{det} A) \cdot (\text{det} B)}\\ &&\\ & = & \displaystyle{0\cdot (\text{det} B)}\\ &&\\ & = & \displaystyle{0.} \end{array}}
Since  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det } C =0,}   we know  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}   is not invertible, which is a contradiction.
So,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   must be invertible and the statement is true.
Final Answer:  
       TRUE

Return to Sample Exam

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