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किसी रैखिक समय-अपरिवर्तनीय निरंतर-समय प्रणाली का प्रणाली आव्यूह \(A\left[ {\begin{array}{*{20}{c}} 0&1\\ { - 4}&{ - 5} \end{array}} \right]\)द्वारा दिया जाता है। अभिलक्षण समीकरण के मूलों के मान क्या होगें?

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  1. -1, -4
  2. -1, -5
  3. -4, -5
  4. 0, -1

Answer (Detailed Solution Below)

Option 1 : -1, -4
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Detailed Solution

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परिकल्पना :

अभिलक्षण समीकरण के मूल के लिए :

[ sI - A ] = 0

जहाँ 

I = इकाई आव्यूह 

\({\rm{I}} = \left[ {\;\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\)

A = दिया गया आव्यूह 

व्याख्या:

दिया है 

\(A=\left[ {\begin{array}{*{20}{c}} 0&1\\ { - 4}&{ - 5} \end{array}} \right]\)

अभिलक्षण समीकरण के मूल के लिए:

[sI - A] = 0

\({\left[ {\left[ {\begin{array}{*{20}{c}} s&0\\ 0&s \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 0&1\\ -4&-5 \end{array}} \right]} \right]^{ }}=0\)

\(\left[ {\begin{array}{*{20}{c}} s&-1\\ { 4}&{ s+ 5} \end{array}} \right]=0\)

s( s + 5 ) + 4 = 0

s2 + 5s + 4 = 0

( s+4 )( s+1 ) = 0

s = -4, -1 

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