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.
