Choose the correct answer.
If x,y,z are nonzero real numbers,then the inverse of matrix
A=\(\begin{bmatrix}x& 0& 0\\ 0& y& 0\\0&0& z\end{bmatrix}\)is
\(\begin{bmatrix}x^{-1}& 0& 0\\ 0& y^{-}1& 0\\0&0& z^{-1}\end{bmatrix}\)
xyz\(\begin{bmatrix}x^{-1}& 0& 0\\ 0& y^{-}1& 0\\0&0& z^{-1}\end{bmatrix}\)
\(\frac{1}{xyz}\begin{bmatrix}x& 0& 0\\ 0& y& 0\\0&0& z\end{bmatrix}\)
\(\frac{1}{xyz}\begin{bmatrix}1& 0& 0\\ 0& 1& 0\\0&0& 1\end{bmatrix}\)
A=\(\begin{bmatrix}x& 0& 0\\ 0& y& 0\\0&0& z\end{bmatrix}\)
\(∴|A|=x(yz-0)=xyz≠0\)
Now,
\(A_{11}=yz,A_{12}=0,A_{13}=0\)
\(A_{21}=0,A_{22}=xz,A_{23}=0\)
\(A_{31}=0,A_{32}=0,A_{33}=xy\)
\(∴adjA\)=\(\begin{bmatrix}yz& 0& 0\\ 0& xz& 0\\0&0& xy\end{bmatrix}\)
\(∴A^{-1}\)=\(\frac{1}{|A|}\)\(adjA\)
=\(\frac{1}{xyz}\begin{bmatrix}yz& 0& 0\\ 0& xz& 0\\0&0& xy\end{bmatrix}\)
=\(\begin{bmatrix}\frac{yz}{xyz}& 0& 0\\ 0& \frac{xz}{xyz}& 0\\0&0& \frac{xy}{xyz}\end{bmatrix}\)
=\(\begin{bmatrix}\frac{1}{x}& 0& 0\\ 0& \frac{1}{y}& 0\\0&0& \frac{1}{z}\end{bmatrix}\)
=\(\begin{bmatrix}x^{-1}& 0& 0\\ 0& y^{-1}& 0\\0&0& z^{-1}\end{bmatrix}\)
The correct answer is A.
A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber.
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place).