Question

If A is a square matrix of order 3 such that adj (2A) = k adj (A), then write the value of k.

Answer 1

$$\begin{gathered} \mathrm{For}\mathrm{any}\mathrm{matrtix}Aof\mathrm{order}n,\mathrm{adj}\left(\lambda A\right)={\lambda }^{n-1}\left(\mathrm{adj}A\right),\mathrm{where}\lambda \mathrm{is}\mathrm{a}\mathrm{constant}. \\ \mathrm{Thus},\mathrm{for}\mathrm{matrix}\mathit{}A\mathrm{of}\mathrm{order}3,\mathrm{we}\mathrm{have} \\ \mathrm{adj}\left(2A\right)=2^{3-1}\left(\mathrm{adj}A\right) \\ \Rightarrow \mathrm{adj}\left(2A\right)=2^2\left(\mathrm{adj}A\right) \\ \Rightarrow \mathrm{adj}\left(2A\right)=4\mathrm{adj}\left(A\right) \\ \Rightarrow k\mathrm{adj}\left(A\right)=4\mathrm{adj}\left(A\right)\left[\because \mathrm{adj}\left(2A\right)=k\mathrm{adj}\left(A\right)\right] \\ \Rightarrow k=4 \end{gathered}$$