Matrices and Determinants 2 Question 40
44. Show that
$ \left|\begin{aligned} & { }^{x} C _r \quad{ }^{x} C _{r+1} \quad{ }^{x} C _{r+2} \\ & { }^{y} C _r \quad{ }^{y} C _{r+1} \quad{ }^{y} C _{r+2} \\ & { }^{z} C _r \quad{ }^{z} C _{r+1} \quad{ }^{z} C _{r+2} \end{aligned}\right| $ $ \left|\begin{aligned} & { }^{x} C _r \quad{ }^{x} C _{r+1} \quad{ }^{x+2} C _{r+2} \\ & { }^{y} C _r \quad{ }^{y} C _{r+1} \quad{ }^{y+2} C _{r+2} \\ & { }^{z} C _r \quad{ }^{z} C _{r+1} \quad{ }^{z+2} C _{r+2} \end{aligned}\right| $
$(1985,3 M)$
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Answer:
Solution:
- Let $\Delta=$ $ \left|\begin{aligned} & { }^{x} C _r \quad{ }^{x} C _{r+1} \quad{ }^{x} C _{r+2} \\ & { }^{y} C _r \quad{ }^{y} C _{r+1} \quad{ }^{y} C _{r+2} \\ & { }^{z} C _r \quad{ }^{z} C _{r+1} \quad{ }^{z} C _{r+2} \end{aligned}\right| $
Applying $C _3 \rightarrow C _3+C _2$
$\Delta=$ $ \left|\begin{aligned} & { }^{x} C _r \quad{ }^{x} C _{r+1} \quad{ }^{x+1} C _{r+2} \\ & { }^{y} C _r \quad{ }^{y} C _{r+1} \quad{ }^{y+1} C _{r+2} \\ & { }^{z} C _r \quad{ }^{z} C _{r+1} \quad{ }^{z+1} C _{r+2} \end{aligned}\right| $
$ \left[\because{ }^{n} C _r+{ }^{n} C _{r-1}={ }^{n+1} C _r\right] $
Applying $C _2 \rightarrow C _2+C _1$
Applying $C _3 \rightarrow C _3+C _2$
Hence proved.
