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Without Expanding, Prove that ∣ ∣ ∣ ∣ a B C X Y Z P Q R ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ X Y Z P Q R a B C ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ Y B Q X a P Z C R ∣ ∣ ∣ ∣ - Mathematics

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Question

Without expanding, prove that

\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]

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Solution

\begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} `R_2 harr R_3=-|(x,y,z),(a,b,c),(p,q,r)| R_1 harrR_2=`\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix}

`|(y,b,q),(x,a,p),(z,c,r)| = |(y,x,z),(b,a,c),(q,p,r)|  C_1 harr C_2=-|(x,y,z),(a,b,c),(p,q,r)|  R_1 harr R_2=`\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix}

Hence proved.

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Chapter 6: Determinants - Exercise 6.2 [Page 61]

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RD Sharma Mathematics [English] Class 12
Chapter 6 Determinants
Exercise 6.2 | Q 46 | Page 61

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