Question

Find the value of the determinant $\left|\begin{array}{ccc}{2}^2& 2^3& 2^4\\ 2^3& 2^4& 2^5\\ 2^4& 2^5& 2^6\end{array}\right|$.

Answer 1

$$\begin{gathered} \left|2^2{2}^3{2}^4 \\ {2}^3{2}^4{2}^5 \\ {2}^4{2}^5{2}^6\right| \\ =2^2\times 2^3\times 2^4\left|122^2 \\ 122^2 \\ 122^2\right|\left[\mathrm{Taking}\mathrm{out}\mathrm{common}\mathrm{factors}\mathrm{from}{R}_1,R_2\mathrm{and}{R}_3\right] \\ =2^2\times 2^3\times 2^4\times 2\left|112^2 \\ 112^2 \\ 112^2\right|=0\left[\mathrm{Two}\mathrm{rows}\mathrm{being}\mathrm{identical}\right] \\ \Rightarrow \left|2^2{2}^3{2}^4 \\ {2}^3{2}^4{2}^5 \\ {2}^4{2}^5{2}^6\right|=0 \end{gathered}$$