Question

Write the value of the determinant \[\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}\]

Answer 1

\[\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix} = p^2 - \left( p + 1 \right)\left( p - 1 \right)\]
\[ = p^2 - \left( p^2 - 1 \right)\]
\[ = p^2 - p^2 + 1\]
\[ = 1\]
Hence, the value of the determinant 

\[\begin{vmatrix}p & p + 1 \\ p - 1 & p\end{vmatrix}\] is 1.