Difference between revisions of "031 Review Part 2, Problem 9"
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<span class="exam">If <math style="vertical-align: 0px">A</math> is an <math style="vertical-align: 0px">n\times n</math> matrix such that <math style="vertical-align: -4px">AA^T=I,</math> what are the possible values of <math style="vertical-align: 0px">\text{det }A?</math> | <span class="exam">If <math style="vertical-align: 0px">A</math> is an <math style="vertical-align: 0px">n\times n</math> matrix such that <math style="vertical-align: -4px">AA^T=I,</math> what are the possible values of <math style="vertical-align: 0px">\text{det }A?</math> | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
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|'''1.''' <math style="vertical-align: -5px">\text{det }(AB)=(\text{det }A)(\text{det }B)</math> | |'''1.''' <math style="vertical-align: -5px">\text{det }(AB)=(\text{det }A)(\text{det }B)</math> | ||
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| − | |'''2.''' <math style="vertical-align: 0px">\text{det }I= | + | |'''2.''' <math style="vertical-align: 0px">\text{det }I=1</math> |
|- | |- | ||
|'''3.''' <math style="vertical-align: 0px">\text{det }A^T=\text{det }A</math> | |'''3.''' <math style="vertical-align: 0px">\text{det }A^T=\text{det }A</math> | ||
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| <math>\text{det }A=\pm 1</math> | | <math>\text{det }A=\pm 1</math> | ||
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| − | [[031_Review_Part_2|'''<u>Return to | + | [[031_Review_Part_2|'''<u>Return to Review Problems</u>''']] |
Latest revision as of 12:40, 15 October 2017
If is an 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 n\times n} matrix such that 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^T=I,} what are the possible values 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 \text{det }A?}
| Foundations: |
|---|
| Recall: |
| 1. 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)} |
| 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 }I=1} |
| 3. 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^T=\text{det }A} |
Solution:
| Step 1: |
|---|
| Using the facts in the Foundations section, 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{1} & = & \displaystyle{\text{det }(I)}\\ &&\\ & = & \displaystyle{\text{det } (AA^T)}\\ &&\\ & = & \displaystyle{(\text{det }A)(\text{det } A^T)}\\ &&\\ & = & \displaystyle{(\text{det }A)(\text{det } A)}\\ &&\\ & = & \displaystyle{(\text{det }A)^2.} \end{array}} |
| Step 2: |
|---|
| Taking the square root of both sides of the equation |
|
| 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 \text{det }A=\pm 1.} |
| 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 \text{det }A=\pm 1} |