If $x, y$ and $z$ are all different from zero and $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z\end{array}\right|=0$, then the value of $x^{-1}+y^{-1}+z^{-1}$ is
The value of $\left|\begin{array}{ccc}x & x+y & x+2 y \\ x+2 y & x & x+y \\ x+y & x+2 y & x\end{array}\right|$ is
If $A$ is a matrix of order $3 \times 3$, then $|3 A|$ is equal to ............. .
If $A$ is a matrix of order $3 \times 3$, then $|3 A|=3 \times 3 \times 3|A|=27|A|$
If $A$ is invertible matrix of order $3 \times 3$, then $\left|A^{-1}\right|$ is equal to ............. .
If $A$ is invertible matrix of order $3 \times 3$, then $\left|A^{-1}\right|=\frac{1}{|A|}$. $\left[\right.$ since, $\left.|A| \cdot\left|A^{-1}\right|=1\right]$
If $x, y, z \in R$, then the value of $\left|\begin{array}{lll}\left(2^x+2^{-x}\right)^2 & \left(2^x-2^{-x}\right)^2 & 1 \\ \left(3^x+3^{-x}\right)^2 & \left(3^x-3^{-x}\right)^2 & 1 \\ \left(4^x+4^{-x}\right)^2 & \left(4^x-4^{-x}\right)^2 & 1\end{array}\right|$ is
We have,
$$\begin{aligned} & \left|\begin{array}{lll} \left(2^x+2^{-x}\right)^2 & \left(2^x-2^{-x}\right)^2 & 1 \\ \left(3^x+3^{-x}\right)^2 & \left(3^x-3^{-x}\right)^2 & 1 \\ \left(4^x+4^{-x}\right)^2 & \left(4^x-4^{-x}\right)^2 & 1 \end{array}\right| \\ & =\left|\begin{array}{lll} \left(2 \cdot 2^x\right)\left(2 \cdot 2^{-x}\right) & \left(2^x-2^{-x}\right)^2 & 1 \\ \left(2 \cdot 3^x\right)\left(2 \cdot 3^{-x}\right) & \left(3^x-3^{-x}\right)^2 & 1 \\ \left(2 \cdot 4^x\right)\left(2 \cdot 4^{-x}\right) & \left(4^x-4^{-x}\right)^2 & 1 \end{array}\right|\quad \begin{array}{r} {\left[\because(a+b)^2-(a-b)^2=4 a b\right]} \\ {\left[\because C_1 \rightarrow C_1-C_2\right]} \end{array} \end{aligned}$$
$=\left|\begin{array}{lll}4 & \left(2^x-2^{-x}\right)^2 & 1 \\ 4 & \left(3^x-3^{-x}\right)^2 & 1 \\ 4 & \left(4^x-4^{-x}\right)^2 & 1\end{array}\right|=0 \quad$ [since, $C_1$ and $C_3$ are proportional to each other]