Prove that 1 1+p 1+p+q

2 3+2p 1+3p+2q = 1

3 6+3p 1+6p+3q

|\begin{matrix} 1 & 1 + p & 1 + p + q \\ 2 & 3 + 2 p & 1 + 3 p + 2 q \\ 3 & 6 + 3 p & 1 + 6 p + 3 q \end{matrix}| = |\begin{matrix} 1 & 1 + p & 1 + p \\ 2 & 3 + 2 p & 1 + 3 p \\ 3 & 6 + 3 p & 1 + 6 p \end{matrix}| + |\begin{matrix} 1 & 1 + p & q \\ 2 & 3 + 2 p & 2 q \\ 3 & 6 + 3 p & 3 q \end{matrix}| B y d e t e r m i n a n t P r o p e r t y = |\begin{matrix} 1 & 1 + p & 1 + p \\ 2 & 3 + 2 p & 1 + 3 p \\ 3 & 6 + 3 p & 1 + 6 p \end{matrix}| + q \begin{matrix}  \end{matrix} |\begin{matrix} 1 & 1 + p & 1 \\ 2 & 3 + 2 p & 2 \\ 3 & 6 + 3 p & 3 \end{matrix}| = |\begin{matrix} 1 & 1 + p & 1 + p \\ 2 & 3 + 2 p & 1 + 3 p \\ 3 & 6 + 3 p & 1 + 6 p \end{matrix}| + \begin{matrix}  \end{matrix} q (0) b y d e t e r m i n a n t p r o p e r t y = |\begin{matrix} 1 & 1 + p & 1 + p \\ 2 & 3 + 2 p & 1 + 3 p \\ 3 & 6 + 3 p & 1 + 6 p \end{matrix}| = |\begin{matrix} 1 & 1 + p & 1 \\ 2 & 3 + 2 p & 1 \\ 3 & 6 + 3 p & 1 \end{matrix}| + \begin{matrix}  \end{matrix} |\begin{matrix} 1 & 1 + p & p \\ 2 & 3 + 2 p & 3 p \\ 3 & 6 + 3 p & 6 p \end{matrix}| = |\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 3 & 6 & 1 \end{matrix}| + |\begin{matrix} 1 & p & 1 \\ 2 & 2 p & 1 \\ 3 & 3 p & 1 \end{matrix}| + \begin{matrix}  \end{matrix} |\begin{matrix} 1 & 1 & p \\ 2 & 3 & 3 p \\ 3 & 6 & 6 p \end{matrix}| + \begin{matrix}  \end{matrix} |\begin{matrix} 1 & p & p \\ 2 & 2 p & 3 p \\ 3 & 3 p & 6 p \end{matrix}| = |\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 3 & 6 & 1 \end{matrix}| + p |\begin{matrix} 1 & 1 & 1 \\ 2 & 2 & 1 \\ 3 & 3 & 1 \end{matrix}| + p \begin{matrix}  \end{matrix} |\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 3 \\ 3 & 6 & 6 \end{matrix}| + p \begin{matrix}  \end{matrix} |\begin{matrix} 1 & 1 & p \\ 2 & 2 & 3 p \\ 3 & 3 & 6 p \end{matrix}| = |\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 3 & 6 & 1 \end{matrix}| + p (0) + p (0) + p (0) = |\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 3 & 6 & 1 \end{matrix}| = 1 (3 \times 1 - 6 \times 1) - 1 (2 \times 1 - 3 \times 1) + 1 (2 \times 6 - 3 \times 3) = 1 (3 - 6) - 1 (2 - 3) + 1 (12 - 9) = - 3 + 1 + 3 = 1 P r o v e d \begin{matrix}  \end{matrix}

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Prove that 1 1+p 1+p+q 2 3+2p 1+3p+2q = 1 3 6+3p 1+6p+3q

Prove that 1 1+p 1+p+q

2 3+2p 1+3p+2q = 1

3 6+3p 1+6p+3q