The value of $\left| {\begin{array}{*{20}{c}}1&{\cos (\beta - \alpha )}&{\cos (\gamma - \alpha )}\\{\cos (\alpha - \beta )}&1&{\cos (\gamma - \beta )}\\{\cos (\alpha - \gamma )}&{\cos (\beta - \gamma )}&1\end{array}} \right|$ is

$A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{array}\right] \Rightarrow A^2-2 A=$

Let $1, \omega$ and $\omega^2$ be the cube roots of unity. If $S$ is the set of all non-singular matrices of the form $M = \begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$ where $a, b, c \in \{\omega, \omega^2\}$,then the number of elements in $S$ is

Let $A$ and $B$ be two symmetric matrices of order $3$.
Statement $-1$: $A(BA)$ and $(AB)A$ are symmetric matrices.
Statement $-2$: $AB$ is a symmetric matrix if the matrix multiplication of $A$ with $B$ is commutative.

Read the following mathematical statements carefully:
$I$. There can exist two triangles such that the sides of one triangle are all less than $1 \text{ cm}$ while the sides of the other triangle are all bigger than $10 \text{ m}$,but the area of the first triangle is larger than the area of the second triangle.
$II$. If $x, y, z$ are all different real numbers,then $\frac{1}{(x - y)^2} + \frac{1}{(y - z)^2} + \frac{1}{(z - x)^2} = \left( \frac{1}{x - y} + \frac{1}{y - z} + \frac{1}{z - x} \right)^2$.
$III$. $\log_3 x \cdot \log_4 x \cdot \log_5 x = (\log_3 x \cdot \log_4 x) + (\log_4 x \cdot \log_5 x) + (\log_5 x \cdot \log_3 x)$ is true for exactly one real value of $x$.
$IV$. $A$ matrix has $12$ elements. The number of possible orders it can have is $6$. Now indicate the correct alternative.

If matrix $A = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$,then: