Determinants Matrices Question 144
Question: The value of the determinant $ \begin{vmatrix} ^{n}{C _{r-1}} & ^{n}C _{r} & (r+1) & ^{n+2}{C _{r+1}} \\ ^{n}C _{r} & ^{n}{C _{r+1}} & (r+2) & ^{n+2}{C _{r+2}} \\ ^{n}{C _{r+1}} & ^{n}{C _{r+2}} & (r+3) & ^{n+2}{C _{r+3}} \\ \end{vmatrix} $ is
Options:
A) $ n^{2}+n-1 $
B) 0
C) $ ^{n+3}{C _{r+3}} $
D) $ ^{n}{C _{r-1}}{{+}^{n}}C _{r}{{+}^{n}}{C _{r+1}} $
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Answer:
Correct Answer: B
Solution:
- [b] $ \Delta = \begin{vmatrix} ^{n}{C _{r-1}} & ^{n}C _{r} & {{(r+1)}^{n+2}}{C _{r+1}} \\ ^{n}C _{r} & ^{n}{C _{r+1}} & {{(r+2)}^{n+2}}{C _{r+2}} \\ ^{n}{C _{r+1}} & ^{n}{C _{r+2}} & {{(r+3)}^{n+2}}{C _{r+3}} \\ \end{vmatrix} $
Applying $ C_1\to C_1+C_2 $ and using $ ^{n}C _{r}=\frac{n}{r}{}^{n-1}{C _{r-1}} $ in $ C_3 $ we get $ \Delta = \begin{vmatrix} ^{n+1}C _{r} & ^{n}C _{r} & (n+2){}^{n+1}C _{r} \\ ^{n+1}{C _{r+1}} & ^{n}{C _{r+1}} & (n+2){}^{n+1}{C _{r+1}} \\ ^{n+1}{C _{r+2}} & ^{n}{C _{r+2}} & (n+2){}^{n+1}{C _{r+2}} \\ \end{vmatrix} $
$ =(n+2) \begin{vmatrix} ^{n+1}C _{r} & ^{n}C _{r} & ^{n+1}C _{r} \\ ^{n+1}{C _{r+1}} & ^{n}{C _{r+1}} & ^{n+1}{C _{r+1}} \\ ^{n+1}{C _{r+2}} & ^{n}{C _{r+2}} & ^{n+1}{C _{r+2}} \\ \end{vmatrix} $ =0 (as $ C_1 $ and $ C_3 $ are identical)
BETA
