If A is an invertible matrix of order 2,then det(A-1) is equal to
det
\(\frac{1}{det(A)}\)
1
0
Since A is an invertible matrix,A-1 exists and A-1=\(\frac{1}{\mid A\mid} adj A.\)
As matrix A is of order 2,let A=\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\).
Then, IAI=ad-bc and adjA=\(\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\).
Now,A-1=\(\frac{1}{\mid A\mid} adj A\)=\(\begin{bmatrix}\frac{d}{\mid A\mid}&\frac{-b}{\mid A \mid}\\ \frac{-c}{\mid A \mid}&\frac{a}{\mid A\mid}\end{bmatrix}\).
therefore IA-1I=\(\begin{vmatrix}\frac{d}{\mid A\mid}&\frac{-b}{\mid A \mid}\\ \frac{-c}{\mid A \mid}&\frac{a}{\mid A\mid}\end{vmatrix}\)
=\(\frac{1}{\mid A \mid^2}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)
=\(\frac{1}{\mid A \mid^2}(ad-bc)=\frac{1}{\mid A \mid^2}.\mid A \mid=\frac{1}{\mid A\mid}\)
\(\therefore\) det(A-1)=\(\frac{1}{det(A)}\)
Hence, the correct answer is B.
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).