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

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2I) - 4(A-I) = O$,where $I$ and $O$ are the identity and null matrices,respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$,where $\alpha, \beta$ and $\gamma$ are real constants,then $\alpha + \beta + \gamma$ is equal to:

Let $a, b, c$ be non-real numbers satisfying the equation $x^5 = 1$ and $S$ be the set of all non-invertible matrices of the form $\begin{bmatrix} 1 & a & b \\ w & 1 & c \\ w^2 & w & 1 \end{bmatrix}$,where $w = e^{\frac{i 2\pi}{5}}$. Then the number of distinct matrices in the set $S$ is:

If $a > 0$ and the discriminant of $ax^2 + 2bx + c$ is negative,then $\left| \begin{array}{ccc} a & b & ax + b \\ b & c & bx + c \\ ax + b & bx + c & 0 \end{array} \right|$ is

If $A$ is a matrix such that $A^2 + A + 2I = O$,then which of the following is $INCORRECT$?

Let $X = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}$. If $f: X \rightarrow \mathbb{R}$ is defined by $f(A) = \det(A)$ for all $A \in X$, then $f$ is