The value of the determinant $\left| \begin{matrix} 0 & x - y & x - z \\ y - x & 0 & y - z \\ z - x & z - y & 0 \end{matrix} \right|$ is:

If $a, b, c > 0$ and $x, y, z \in R$,then the value of the determinant $\left| \begin{array}{ccc} (a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 \\ (b^y + b^{-y})^2 & (b^y - b^{-y})^2 & 1 \\ (c^z + c^{-z})^2 & (c^z - c^{-z})^2 & 1 \end{array} \right|$ is equal to:

Consider the following statements :
$(a)$ If any two rows or columns of a determinant are identical,then the value of the determinant is zero.
$(b)$ If the corresponding rows and columns of a determinant are interchanged,then the value of the determinant does not change.
$(c)$ If any two rows (or columns) of a determinant are interchanged,then the value of the determinant changes in sign.
Which of these are correct?

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:

The determinant $\left|\begin{array}{ccc}a^{2}+10 & a b & a c \\ a b & b^{2}+10 & b c \\ a c & b c & c^{2}+10\end{array}\right|$ is

Evaluate the determinant: $\left| \begin{array}{ccc} a_1 & m a_1 & b_1 \\ a_2 & m a_2 & b_2 \\ a_3 & m a_3 & b_3 \end{array} \right|$