Difference between revisions of "031 Review Part 2, Problem 8"

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1:    
 
!Step 1:    
 +
|-
 +
|We begin by augmenting the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; with the identity matrix. Hence, we get
 
|-
 
|-
 
|
 
|
 +
::<math>\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 8 & 1 & 0 & 0\\
 +
          2 & 4  & 11 & 0 & 1 & 0\\
 +
          1 & 2 & 5 & 0 & 0 & 1
 +
        \end{array}\right].</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|Now, we row reduce the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; to obtain the identity matrix. Hence, we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 8 & 1 & 0 & 0\\
 +
          2 & 4  & 11 & 0 & 1 & 0\\
 +
          1 & 2 & 5 & 0 & 0 & 1
 +
        \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 8 & 1 & 0 & 0\\
 +
          0 & -2  & -5 & -2 & 1 & 0\\
 +
          0 & -1 & -3 & -1 & 0 & 1
 +
        \end{array}\right]}\\
 +
&&\\
 +
& \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 8 & 1 & 0 & 0\\
 +
          0 & 1  & 3 & 1 & 0 & -1\\
 +
          0 & -2 & -5 & -2 & 1 & 0
 +
        \end{array}\right]}\\
 +
&&\\
 +
& \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 8 & 1 & 0 & 0\\
 +
          0 & 1  & 3 & 1 & 0 & -1\\
 +
          0 & 0 & 1 & 0 & 1 & -1
 +
        \end{array}\right]}\\
 +
&&\\
 +
& \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 3 & 0 & 1 & -8 & 8\\
 +
          0 & 1  & 0 & 1 & -3 & 2\\
 +
          0 & 0 & 1 & 0 & 1 & -1
 +
        \end{array}\right]}\\
 +
&&\\
 +
& \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 
 +
          1 & 0 & 0 & -2 & 1 & 2\\
 +
          0 & 1  & 0 & 1 & -3 & 2\\
 +
          0 & 0 & 1 & 0 & 1 & -1
 +
        \end{array}\right].}
 +
\end{array}</math>
 +
|-
 +
|Therefore, the inverse of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\left[\begin{array}{ccc} 
 +
          -2 & 1 & 2\\
 +
            1 & -3 & 2\\
 +
            0 & 1 & -1
 +
        \end{array}\right]</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; &nbsp; &nbsp;  
+
|&nbsp;&nbsp; &nbsp; &nbsp; <math>A^{-1}=\left[\begin{array}{ccc} 
 +
          -2 & 1 & 2\\
 +
            1 & -3 & 2\\
 +
            0 & 1 & -1
 +
        \end{array}\right]</math>
 
|}
 
|}
 
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 20:19, 10 October 2017

Let    Find  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^{-1}}   if possible.


Foundations:  
To find the inverse of a matrix  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,}   you augment the matrix  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}  
with the identity matrix and row reduce  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}   to the identity matrix.


Solution:

Step 1:  
We begin by augmenting the matrix  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}   with the identity matrix. Hence, we get
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 \left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 2 & 4 & 11 & 0 & 1 & 0\\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right].}
Step 2:  
Now, we row reduce the matrix  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}   to obtain the identity matrix. Hence, 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{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 2 & 4 & 11 & 0 & 1 & 0\\ 1 & 2 & 5 & 0 & 0 & 1 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & -2 & -5 & -2 & 1 & 0\\ 0 & -1 & -3 & -1 & 0 & 1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & 1 & 3 & 1 & 0 & -1\\ 0 & -2 & -5 & -2 & 1 & 0 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 8 & 1 & 0 & 0\\ 0 & 1 & 3 & 1 & 0 & -1\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 3 & 0 & 1 & -8 & 8\\ 0 & 1 & 0 & 1 & -3 & 2\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & -2 & 1 & 2\\ 0 & 1 & 0 & 1 & -3 & 2\\ 0 & 0 & 1 & 0 & 1 & -1 \end{array}\right].} \end{array}}

Therefore, the inverse of  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
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 \left[\begin{array}{ccc} -2 & 1 & 2\\ 1 & -3 & 2\\ 0 & 1 & -1 \end{array}\right]}


Final Answer:  
       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^{-1}=\left[\begin{array}{ccc} -2 & 1 & 2\\ 1 & -3 & 2\\ 0 & 1 & -1 \end{array}\right]}

Return to Sample Exam

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