If $\Delta = \begin{vmatrix} x+y+z^2 & x^2+y+z & x+y^2+z \\ z^2 & x^2 & y^2 \\ x+y & y+z & x+z \end{vmatrix}$,(where $x \neq y \neq z$ and $x, y, z \in \mathbb{R} - \{0\}$),then $\Delta = $ . . . . . . .

Show that $\Delta=\left|\begin{array}{ccc} (y+z)^{2} & x y & z x \\ x y & (x+z)^{2} & y z \\ x z & y z & (x+y)^{2} \end{array}\right|=2 x y z(x+y+z)^{3}$

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:

$\left| {\begin{array}{ccc} 1 & 1+ac & 1+bc \\ 1 & 1+ad & 1+bd \\ 1 & 1+ae & 1+be \end{array}} \right| = $

If ${a_1}, {a_2}, {a_3}, \dots, {a_n}, \dots$ are in $G.P.$,then the value of the determinant $\left| \begin{array}{ccc} \log {a_n} & \log {a_{n+1}} & \log {a_{n+2}} \\ \log {a_{n+3}} & \log {a_{n+4}} & \log {a_{n+5}} \\ \log {a_{n+6}} & \log {a_{n+7}} & \log {a_{n+8}} \end{array} \right|$ 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?