How to Find eigen values and eigen vectors for a matrix?

Q: How to Find eigen values and eigen vectors for a matrix?
A: This topic is very important for GATE, as one question is asked almost every year in which you are required to find out the eigen vector of a matrix.
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a) Eigen Value:
Suppose a matrix  "P" is given. Now, eigen values can be found by simply solving the expression [P-λI]=0; for "λ", where I is identity matrix.
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For example:     P=  5    3
                              1    3
Now, doing [P-λI]=0, we get,    (5-λ)(3-λ)-3=0,
which gives λ=6,2
Now, these are eigen values of this matrix.
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b) Eigen Vector:
Now, in order to find out Eigen vector, select any one of the eigen values.

Now use the equation    [P][x      y]'     =    λ[x     y]'  

to find out the ratio of x and y.

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For example, we select λ=2.
Now, using the equation, we get    5x+3y=2x  and  x+3y=2y;
From each of the equations,  we get    x+y=0 or x=-y;
Hence, ratio of x and y, i.e. [1   -1] is our eigen vector..
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.Remember: Both the equations in second last step should give same eigen vector.

Queuing Theory Formulas

Important formulas for Queuing Theory(Acc. to GATE requirement):

No. of People in the system,    Ns =   λ  / (μ- λ )


No. of People in Queue,             Nq = Ls - λ  / μ =  λ  / (μ- λ ) -   λ  / μ
                                                    =  λ ^2 / μ(μ- λ )


Average waiting time in system,  Ws = Ns/ λ  = 1/(μ- λ )


Average waiting time in queue,    Wq = Nq/ λ  =  λ / μ(μ- λ )


Complete Queuing Theory is quite versatile but this much of formulas are sufficient for GATE

We know a reversible adiabatic process is an isentropic process......but  is an isentropic process necessarily a reversible adiabatic process. Why or why not?

Types of fluids

The Hershel- Buckley fluids are known as Thixotropic fluid, while Bingham fluids are know as Ideal Plastic fluids.

                          General equation is      τ =     μ  .[] ^ n  +  τ0 

1)     For Bingham and Thixotropic fluids, τ0 is greater than zero.

2)     For all other fluids, τ0 = 0

3)     For newtonian fluids, τ0  is zero and value of exponent n is 1.

4)     For Pseudoplastic and Thixotropic fluids, n<1.

5)     For Dilatant fluids, n> 1.

Equal amounts of Water at 80 deg C and water at 0 deg C are mixed together. What would be the final temperature of mixture? What will be change in answer if water at 0 deg C is in form of ice?

There are two power-plants, Nuclear power plant and Thermal Power Plant. Using same conditions, which of these would be having more efficiency?

Hint: Producing Nuclear energy, energy of 2 out of 3 Neutrons is absorbed during chain reaction.

What is the effect on power generated, efficiency when we make a boiler super critical?