If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}?$

Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ vanish?

1. a + b + c = 0

2. a³ + b³ + c³ = 3abc

3. a² + b² + c² - ab - bc - ca = 0

Select the correct answer using the code given below:

Consider the following in respect of the matrices:

A = [m n], B = [-n -m] & $ C = \begin{bmatrix} m \\ -m\end{bmatrix} $

 

1. CA = CB

2. AC = BC

3. C(A + B) = CA + CB

Which of the above statements is/are correct?

If $A = \begin{bmatrix} 2 \sin \theta & \cos \theta & 0 \\ -2\cos \theta & \sin \theta & 0 \\ -1 & 1 & 1 \end{bmatrix},$ then what is A(adj A) equal to?

(where) I is the identity matrix.