Matrices And Determinants Ques 7
- For all values of $A, B, C$ and $P, Q, R$, show that
(1994, 4M)
$ \left|\begin{array}{ccc} \cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R) \end{array}\right|=0 $
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Solution:
Let $\quad \Delta=\left|\begin{array}{lll}\cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R)\end{array}\right|$
$ \begin{aligned} & \Rightarrow \quad \Delta=\left| \begin{array}{ll} \cos A \cos P+\sin A \sin P & \cos (A-Q) & \cos (A-R) \\ \cos B \cos P+\sin B \sin P & \cos (B-Q) & \cos (B-R) \\ \cos C \cos P+\sin C \sin P & \cos (C-Q)& \cos (C-R) \end{array}\right| \\ & \Rightarrow \quad \Delta=\left|\begin{array}{lll} \cos A \cos P & \cos (A-Q) & \cos (A-R) \\ \cos B \cos P & \cos (B-Q) & \cos (B-R) \\ \cos C \cos P & \cos (C-Q) & \cos (C-R) \end{array}\right| \\ & \quad \quad \quad +\left|\begin{array}{lll} \sin A \sin P & \cos (A-Q) & \cos (A-R) \\ \sin B \sin P & \cos (B-Q) & \cos (B-R) \\ \sin C \sin P & \cos (C-Q) & \cos (C-R) \end{array}\right| \\ & \Rightarrow \Delta=\cos P\left|\begin{array}{ccc} \cos A & \cos (A-Q) & \cos (A-R) \\ \cos B & \cos (B-Q) & \cos (B-R) \\ \cos C & \cos (C-Q) & \cos (C-R) \end{array}\right| \\ & \quad \quad \quad +\sin P\left|\begin{array}{lll} \sin A & \cos (A-Q) & \cos (A-R) \\ \sin B & \cos (B-Q) & \cos (B-R) \\ \sin C & \cos (C-Q) & \cos (C-R) \end{array}\right| \\ \end{aligned} $
Applying $C_2 \rightarrow C_2-C_1 \cos Q, C_8 \rightarrow C_8-C_1 \cos R$ in first determinant and $C_2 \rightarrow C_2-C_1 \sin Q$ and in second determinant
$ \begin{aligned} & \Rightarrow \Delta=\cos P\left|\begin{array}{lll} \cos A & \sin A \sin Q & \sin A \sin R \\ \cos B & \sin B \sin Q & \sin B \sin R \\ \cos C & \sin C \sin Q & \sin C \sin R \end{array}\right| \\ & +\sin P\left|\begin{array}{lll} \sin A & \cos A \cos Q & \cos A \cos R \\ \sin B & \cos B \cos Q & \cos B \cos R \\ \sin C & \cos C \cos Q & \cos C \cos R \end{array}\right| \\ & \Delta=\cos P \sin Q \sin R\left|\begin{array}{lll} \cos A & \sin A & \sin A \\ \cos B & \sin B & \sin B \\ \cos C & \sin C & \sin C \end{array}\right| \\ & +\sin P \cos Q \cos R\left|\begin{array}{lll} \sin A & \cos A & \cos A \\ \sin B & \cos B & \cos B \\ \sin C & \cos C & \cos C \end{array}\right| \\ & \Delta=0+0=0 \\ \end{aligned} $
BETA
