निम्नलिखित आव्यूह के लिए det(3A) का मान ज्ञात कीजिए।

\({\rm{A}} = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 4&7&1\\ { - 1}&3&2\\ { - 2}&0&5 \end{array}} \right]\)

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  1. 1458
  2. 81
  3. 27
  4. 1971

Answer (Detailed Solution Below)

Option 4 : 1971
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Detailed Solution

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संकल्पना:

1. एक 3 × 3 आव्यूह का सारणिक:

  • माना कि A एक 3 × 3 आव्यूह निम्न दिया गया है:

\({\rm{A}} = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} {\rm{a}}&{\rm{b}}&{\rm{c}}\\ {\rm{f}}&{\rm{e}}&{\rm{d}}\\ {\rm{g}}&{\rm{h}}&{\rm{i}} \end{array}} \right]\)

तो |A| के मान को det(A) के रूप में भी लिखा गया है:

det (A) = a (ei - dh) – b (fi - dg) + c (fh - eg)

2. एक आव्यूह की सारणिक का गुण:

  • माना कि A कोटि n × n और det(A) = k वाला एक आव्यूह है। तो एक सदिश c के लिए निम्नलिखित गुण हैं:

          det(cA) = cn det(A)

गणना:

सर्वप्रथम दिए गए आव्यूह की सारणिक का मूल्यांकन कीजिए:

det(A) = 4(15 - 0) – 7(-5 + 4) + 1(0 + 6)

= 4(15) -7(-1) + 1(6)

= 60 + 7 + 6

= 73

अब गुण का प्रयोग करने पर det(3A) का मान निम्न है:

det(3A) = 33 det(A)

= 27 × 73

= 1971

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