Let the determinant of a $3 \times 3$ matrix $A$ be $6$. If $B$ is a matrix defined by $B = 5A^2$,then the determinant of $B$ is:

If $a, b, c$ are all different and $\left| \begin{array}{ccc} a & a^3 & a^4 - 1 \\ b & b^3 & b^4 - 1 \\ c & c^3 & c^4 - 1 \end{array} \right| = 0$,then the value of $abc(ab + bc + ca)$ is

If $A = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 3 \\ 4 & 1 \end{bmatrix}$,then which of the following properties holds true?

Let $a-2b+c=1$. If $f(x) = \begin{vmatrix} x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3 \end{vmatrix}$,then:

If $\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3\end{array}\right|=0$ and $x \neq y \neq z$, then $1+x y z$ is equal to

Let $A = \begin{bmatrix} 1 + x^2 - y^2 - z^2 & 2(xy + z) & 2(zx - y) \\ 2(xy - z) & 1 + y^2 - z^2 - x^2 & 2(yz + x) \\ 2(zx + y) & 2(yz - x) & 1 + z^2 - x^2 - y^2 \end{bmatrix}$. Then $\det(A)$ is equal to: