यदि A2 - 2A - I = 0 है, तो A का व्युत्क्रम _____ है। 

  1. I
  2. A + 2
  3. A - 2
  4. A

Answer (Detailed Solution Below)

Option 3 : A - 2
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Detailed Solution

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अवधारणा:

आव्यूह व्युत्क्रम के गुण:

यदि A और B व्युत्क्रमणीय आव्यूह हैं, तो व्युत्क्रम आव्यूह में निम्नलिखित गुण होने चाहिए:

  • (AB) - 1 = B - 1 A - 1
  • (A - 1) - 1 = A
  • (AT) - 1 = (A - 1)T
  • (KA -1 ) =  किसी भी K ≠ 0 के लिए \(\rm \frac{1}{k}\;{A^{ - 1}}\) 
  • (An) - 1 = (A - 1)n
  • AA - 1 = A - 1A = I

गणना:

दिया गया है: A2 - 2A - I = 0

⇒ A.A - 2A = I

A-1 से गुणा करने पर हमें प्राप्त होता है

⇒ AAA-1 - 2AA-1 = IA-1

⇒ AI - 2I = A-1             [∵ AA - 1 = A - 1A = I]

∴ A-1 = A - 2

A का व्युत्क्रम A - 2 है

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