If $a, b, c$ are three complex numbers such that $a^2 + b^2 + c^2 = 0$ and $\begin{vmatrix} (b^2 + c^2) & ab & ac \\ ab & (c^2 + a^2) & bc \\ ac & bc & (a^2 + b^2) \end{vmatrix} = K a^2 b^2 c^2$,then the value of $K$ is:

If the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1 \end{bmatrix}$ satisfies the equation $A^{20} + \alpha A^{19} + \beta A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ for some real numbers $\alpha$ and $\beta$,then $\beta - \alpha$ is equal to ........ .

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ satisfies the equation $x^2 - (a + d)x + k = 0$,then

Suppose $A$ is a $3 \times 3$ matrix consisting of integer entries that are chosen at random from the set $\{-1000, -999, \ldots, 999, 1000\}$. Let $P$ be the probability that either $A^2 = -I$ or $A$ is diagonal,where $I$ is the $3 \times 3$ identity matrix. Then,

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = [p_{ij}]$ be a $2 \times 2$ matrix with $p_{ij} = \omega^{i+j}$. For $P^2 \neq 0$,if $P^k = P$,then $k$ is equal to

In $\triangle ABC$,if $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$,then $\cos A \cos B+\cos B \cos C+\cos C \cos A=$