Question

The integral value of $\lambda$ for which vectors ${\lambda }^2\hat{i}-2\hat{j},2{\lambda }^2\hat{j}+64\hat{k},-{\lambda }^2\hat{k}-\hat{i}$ are linearly dependent, is equal to

Answer 1

Given : ${\lambda }^2\hat{i}-2\hat{j},2{\lambda }^2\hat{j}+64\hat{k},-{\lambda }^2\hat{k}-\hat{i}$ are linearly dependent
$\Rightarrow \left|\begin{array}{ccc}{\lambda }^2& -2& 0\\ 0& 2{\lambda }^2& 64\\ -1& 0& -{\lambda }^2\end{array}\right|=0\Rightarrow {\lambda }^2(-2{\lambda }^4-0)+2(0+64)+0=0\Rightarrow {\lambda }^6=64\Rightarrow \lambda =2$