For the matrix A=\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\),find the numbers a and b such that A2+ aA+bI=O.
Let A=\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\)
A2=\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\)\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\)=\(\begin{bmatrix}9+2&6+2\\3+1&2+1\end{bmatrix}\)
=\(\begin{bmatrix}11&8\\4&3\end{bmatrix}\)
Now, A2+aA+bI=0 [post multiplying by A-1 as IAI≠0]
\(\Rightarrow\) (AA)A-1+aAA-1+bAA-1=0
\(\Rightarrow\) A(AA-1)+aI+b(IA-1)=0
AI+aI+bA-1=0
\(\Rightarrow\) A+aI=-bA-1
\(\Rightarrow\) A-1=-\(\frac{1}{b}\)(A+aI)
Now, A-1=\(\frac{1}{\mid A\mid}\)adj A=\(\frac{1}{1}\)\(\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\)
we have:
\(\begin{bmatrix}1&-2\\-1&3\end{bmatrix}\)=-\(\frac{1}{b}\)\(\bigg(\)\(\begin{bmatrix}3&2\\1&1\end{bmatrix}\)+\(\begin{bmatrix}a&0\\0&a\end{bmatrix}\)\(\bigg)\)=-\(\frac{1}{b}\)\(\begin{bmatrix}3+a&2\\1&1+a\end{bmatrix}\)=\(\begin{bmatrix}\frac{-3-a}{b}&\frac{-2][b}2\\\frac{-1}{b}&\frac{-1-a}{b}\end{bmatrix}\)
Comparing the corresponding elements of the two matrices have:
-\(\frac{1}{b}=-1\geq b=1\)
\(\frac{-3-a}{b}=1\geq -3-a=1\)
\(\Rightarrow\) a=-4
Hence, −4 and 1 are the required values of a and b respectively.
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).