Determinants

If $A$ is a $3\times 3$ matrix with $\det(A)=3$, then $\det(\operatorname{adj}A)$ is:

If $ a,b,c $ are coplanar, evaluate $ [,2a - b, 2b - c,2c - a,] $

If $\omega$ is a cube root of unity, then find the value of determinant $\begin{vmatrix} 1+\omega &\omega^{2} &-\omega \\ 1+\omega^{2}&\omega &-\omega^{2} \\ \omega^{2}+\omega&\omega &-\omega^{2} \end{vmatrix}$

If the matrix $ \begin{bmatrix} -1 & 3 & 2 \\ 1& k &-3 \\ 1 & 4 & 5\\ \end{bmatrix}$ has an inverse matrix, then the value of K is:

If $\vec{a}=\hat{i}-\hat{k},\, \vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k}$ and $\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}$ , then $[\vec{a} , \vec{b}, \vec{c}]$ depends on