Difference between revisions of "031 Review Part 2, Problem 5"

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(Created page with "<span class="exam">Consider the matrix  <math style="vertical-align: -31px">A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\...")
 
 
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<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=  
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<span class="exam">Let &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; be &nbsp;<math style="vertical-align: 0px">6\times 6</math>&nbsp; matrices with &nbsp;<math style="vertical-align: -1px">\text{det }A=-10</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{det }B=5.</math>&nbsp; Use properties of determinants to compute:
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    \begin{bmatrix}
 
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          1 & -4 & 9 & -7 \\
 
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          -1 & 2  & -4 & 1 \\
 
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          5 & -6 & 10 & 7
 
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        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
 
  
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::<math>B=   
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<span class="exam">(a) &nbsp;<math style="vertical-align: -2px">\text{det }3A</math>
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    \begin{bmatrix}
 
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          1 & 0 & -1 & 5 \\
 
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          0 & -2  & 5 & -6 \\
 
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          0 & 0 & 0 & 0
 
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        \end{bmatrix}.</math>     
 
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<span class="exam">(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
 
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<span class="exam">(b) Find bases for &nbsp;<math style="vertical-align: 0px">\text{Col }A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>&nbsp; Find an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: -5px">\text{Col }A,</math>&nbsp; as well as an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>
 
  
 +
<span class="exam">(b) &nbsp;<math style="vertical-align: -7px">\text{det }(A^TB^{-1})</math>
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 +
|-
 +
|Recall:
 +
|-
 +
|'''1.''' If the matrix &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; is identical to the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; except the entries in one of the rows of &nbsp;<math style="vertical-align: 0px">B</math>&nbsp;
 +
|-
 +
|
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:are each equal to the corresponding entries of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; multiplied by the same scalar &nbsp;<math style="vertical-align: -4px">c,</math>&nbsp; then
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|-
 +
|
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::<math>\text{det }B=c(\text{det }A).</math> 
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|-
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|'''2.''' &nbsp;<math style="vertical-align: -5px">\text{det } (AB)=(\text{det }A)(\text{det }B)</math>
 +
|-
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|'''3.''' For an invertible matrix &nbsp;<math style="vertical-align: -4px">A,</math>&nbsp; since &nbsp;<math style="vertical-align: 0px">AA^{-1}=I</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">\text{det }I=1,</math>&nbsp; we have
 
|-
 
|-
 
|
 
|
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::<math>\text{det }A^{-1}=\frac{1}{\text{det } A}.</math>
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|
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|Every entry of the matrix &nbsp;<math style="vertical-align: 0px">3A</math>&nbsp; is &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; times the corresponding entry of &nbsp;<math style="vertical-align: 0px">A.</math>
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|-
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|So, we multiply every row of the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; by &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; to get &nbsp;<math style="vertical-align: 0px">3A.</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;
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|-
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|Hence, we have
 
|-
 
|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{\text{det }(3A)} & = & \displaystyle{3^6(\text{det }A)}\\
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&&\\
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& = & \displaystyle{3^6 (-10)}\\
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&&\\
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& = & \displaystyle{-7290.}
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\end{array}</math>
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Using properties of determinants, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{\text{det }(A^TB^{-1})} & = & \displaystyle{\text{det }(A^T)\cdot \text{det }(B^{-1})}\\
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&&\\
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& = & \displaystyle{\text{det }(A)\cdot \text{det }(B^{-1}).}
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\end{array}</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;
 +
|-
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|Continuing, we obtain
 
|-
 
|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{\text{det }(A^TB^{-1})} & = & \displaystyle{\text{det }(A) \cdot \frac{1}{\text{det }(B)}}\\
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&&\\
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& = & \displaystyle{(-10)\cdot \frac{1}{5}}\\
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&&\\
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& = & \displaystyle{-2.}
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\end{array}</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>\text{det }(3A)=-7290</math>
 
|-
 
|-
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math>\text{det }(A^TB^{-1})=-2</math>
 
|}
 
|}
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[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 12:29, 15 October 2017

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 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}   be  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 6\times 6}   matrices with  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=-10}   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 \text{det }B=5.}   Use properties of determinants to compute:

(a)  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 }3A}

(b)  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^TB^{-1})}

Foundations:  
Recall:
1. If 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 B}   is identical to 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}   except the entries in one of the rows 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 B}  
are each equal to the corresponding entries 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}   multiplied by the same scalar  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,}   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 }B=c(\text{det }A).}
2.  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 } (AB)=(\text{det }A)(\text{det }B)}
3. For an invertible 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,}   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 AA^{-1}=I}   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 \text{det }I=1,}   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 \text{det }A^{-1}=\frac{1}{\text{det } A}.}


Solution:

(a)

Step 1:  
Every entry of 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 3A}   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 3}   times the corresponding entry of  
So, we multiply every row of 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}   by  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 3}   to 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 3A.}
Step 2:  
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{\text{det }(3A)} & = & \displaystyle{3^6(\text{det }A)}\\ &&\\ & = & \displaystyle{3^6 (-10)}\\ &&\\ & = & \displaystyle{-7290.} \end{array}}

(b)

Step 1:  
Using properties of determinants, 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 }(A^TB^{-1})} & = & \displaystyle{\text{det }(A^T)\cdot \text{det }(B^{-1})}\\ &&\\ & = & \displaystyle{\text{det }(A)\cdot \text{det }(B^{-1}).} \end{array}}

Step 2:  
Continuing, we obtain

       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 }(A^TB^{-1})} & = & \displaystyle{\text{det }(A) \cdot \frac{1}{\text{det }(B)}}\\ &&\\ & = & \displaystyle{(-10)\cdot \frac{1}{5}}\\ &&\\ & = & \displaystyle{-2.} \end{array}}


Final Answer:  
   (a)     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 }(3A)=-7290}
   (b)     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^TB^{-1})=-2}

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