Question

Evaluate

\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .\]

Answer 1

\[Let A = \begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix} = 2 \left( 204 - 100 \right) - 3 \left( 156 - 75 \right) + 7 \left( 260 - 255 \right)\]
\[ \Rightarrow A = 2\left( 104 \right) - 3\left( 81 \right) + 7\left( 5 \right)\]
\[ \Rightarrow A = 208 - 243 + 35\]
\[ \Rightarrow A = 243 - 243 = 0\]
\[\]
\[ \therefore \begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix} = 0\]
\[\]
\[ \Rightarrow \begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 = 0^2 = 0 \left[ \because \det A^2 = \left( \det A \right)^2 \right]\]
\[\]