Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:

Let $B=\begin{bmatrix} 1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4 \end{bmatrix}, \alpha > 2$ be the adjoint of a matrix $A$ and $|A|=2$. Then the value of $\begin{bmatrix} \alpha & -2\alpha & \alpha \end{bmatrix} B \begin{bmatrix} \alpha \\ -2\alpha \\ \alpha \end{bmatrix}$ is equal to:

For a square matrix $B$ of order $3$,if $B^T=B^{-1}$ and $|B|=1$,then $|B-I|=$

If the matrix $A=\begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$ satisfies $A(A^{3}+3I)=2I$,then the value of $K$ is:

Let $A^{-1} = \begin{bmatrix} 1 & 2017 & 2 \\ 1 & 2017 & 4 \\ 1 & 2018 & 8 \end{bmatrix}$. Then,$|2A| - |2A^{-1}|$ is equal to

Let $\omega = - \frac{1}{2} + i\frac{\sqrt{3}}{2}$. Then the value of the determinant $\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 - \omega^2 & \omega^2 \\ 1 & \omega^2 & \omega^4 \end{array} \right|$ is