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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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