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If a1, a2, a3, _ _ _ _ _, a9 are in GP, then what is the value of the following determinant?

\(\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|\)

1 Answer

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Best answer
Correct Answer - Option 1 : 0

Concept:

Let us consider sequence a1, a2, a3 …. an is a G.P.

So, common ratio = \(\rm r = \frac{a_{p + 1}}{a_{p}} \), Where p ∈ Positive Integer

Logarithmic formula: log a - log b = \(\rm \frac{log a}{log b}\)

Properties of Determinants:

  • Determinant evaluated across any row or column is same.
  • If all the elements of a row or column are zeroes, then the value of the determinant is zero.
  • The determinant of an Identity matrix is 1.
  • If rows and columns are interchanged then the value of the determinant remains the same (value does not change).
  • If any two-row or two-column of a determinant are interchanged the value of the determinant is multiplied by -1.
  • If two rows or two columns of a determinant are identical the value of the determinant is zero.

 

Calculation:

Let r be the common ratio of the given GP

Now,

\(\left| {\begin{array}{*{20}{c}} {{ln\:a_1}}&{{ln\:a_2}}&{{ln\:a_3}}\\ {{ln\:a_4}}&{{ln\:a_5}}&{{ln\:a_6}}\\ {{ln\:a_7}}&{{ln\:a_8}}&{{ln\:a_9}} \end{array}} \right|\)

Apply, C2 →  C2 - Cand C3→ C3 - C1

\(\left| {\begin{array}{*{20}{c}} {{\ln\:a_1}}&{{\ln\:a_2 - \ln a_1}}&{{\ln\:a_3 - \ln a _2}}\\ {{\ln\:a_4}}&{{\ln\:a_5 - \ln a_4}}&{{\rm ln\:a_6 - \ln a_5}}\\ {{\rm ln\:a_7}}&{\rm{ln\:a_8 - \ln a_7}}&{\rm{ln\:a_9 -\ln a_8}} \end{array}} \right|\)

\(\left| {\begin{array}{*{20}{c}} {{\ln\:a_1}}&{{\ln r}}&{{\ln r }}\\ {{\ln\:a_4}}&{{\ln r }}&{{\ln r}}\\ {{\ln\:a_7}}&{{\ln r }}&{{\ln r}} \end{array}} \right|\)      [Since \(\rm \frac{a_{p + 1}}{a_{p}} = r\) and log ap + 1 - log a= \(\rm log \frac{a_{p + 1}}{a_{p}} = log r\)]

= 0                                  [Since C3 = C2]

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