Question

Find the value of the determinant \[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix}\].

Answer 1

\[\begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix} \] 
\[ = 2^2 \times 2^3 \times 2^4 \begin{vmatrix} 1 & 2 & 2^2 \\1 & 2 & 2^2 \\1 & 2 & 2^2 \end{vmatrix} \left[\text{ Taking out common factors from }R_1 , R_2\text{ and }R_3 \right]\] 
\[ = 2^2 \times 2^3 \times 2^4 \times 2 \begin{vmatrix} 1 & 1 & 2^2 \\1 & 1 & 2^2 \\1 & 1 & 2^2 \end{vmatrix} = 0 \left[\text{ Two rows being identical }\right]\] 
\[ \Rightarrow \begin{vmatrix}2^2 & 2^3 & 2^4 \\ 2^3 & 2^4 & 2^5 \\ 2^4 & 2^5 & 2^6\end{vmatrix} = 0\]