If $\left| \begin{array}{ccc} a & b & c \\ m & n & p \\ x & y & z \end{array} \right| = k$,then $\left| \begin{array}{ccc} 6a & 2b & 2c \\ 3m & n & p \\ 3x & y & z \end{array} \right| = $

If $x, y \in R$ and $\left|\begin{array}{lll}\left(a^x+a^{-x}\right)^2 & \left(a^x-a^{-x}\right)^2 & 1 \\ \left(b^x+b^{-x}\right)^2 & \left(b^x-b^{-x}\right)^2 & 1 \\ \left(c^x+c^{-x}\right)^2 & \left(c^x-c^{-x}\right)^2 & 1\end{array}\right| = 2y+6$,then $y=$

The value of $\left| \begin{array}{ccc} 441 & 442 & 443 \\ 445 & 446 & 447 \\ 449 & 450 & 451 \end{array} \right|$ is

Let $A$ be a $3 \times 3$ matrix with $\operatorname{det}(A) = 4$. Let $R_{i}$ denote the $i^{\text{th}}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R_{2} \rightarrow 2R_{2} + 5R_{3}$ on $2A$,then $\operatorname{det}(B)$ is equal to:

By using properties of determinants,show that:
$\left|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)$

If $A$ is a square matrix of order $n$ and $A = kB$,where $k$ is a scalar,then $|A|=$