TIFR CSE 2019 | Part A | Question: 15

Question

Consider the matrix

$$A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$$

What is $\displaystyle \lim_{n→\infty}$$A^n$ ?

• A. $\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$
• B. $\begin{bmatrix} \frac{1}{4} &\frac{1}{2} & \frac{1}{2}\\ \frac{1}{4}& \frac{1}{2} & \frac{1}{2}\\ \frac{1}{4}& \frac{1}{2} & \frac{1}{2}\end{bmatrix}$
• C. $\begin{bmatrix} \frac{1}{2} &\frac{1}{4} & \frac{1}{4}\\ \frac{1}{2}& \frac{1}{4} & \frac{1}{4}\\ \frac{1}{2}& \frac{1}{4} & \frac{1}{4}\end{bmatrix}$
• D. $\begin{bmatrix} 0 &\frac{1}{2} & \frac{1}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\end{bmatrix}$
• E. $\text{The limit exists, but it is none of the above}$

Comments

Consider this approach.Let A^inf=X

Now

A^inf=A*A^inf=A^inf*A

So

X=A*X=X*A

Substitute for X from options

Now B and C are not possible

oly A and D are possible.

Still cant eliminate A.

@srestha

Answer 1

We observe that given matrix A is a right stochastic matrix. That is, for every row the row sum is 1. What does that mean? It means the given matrix corresponds to some Markov system. which is: A 3 state system as follows : (with each edge being the probability of the system going from a state to other.)

 

Now we find the steady-state probability vector as follows :

[$x_1 , x_2, x_3$] $\begin{bmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\\\ 0 & \frac{3}{4} & \frac{1}{4} \\ \\ 0 & \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ = [$x_1, x_2, x_3$]

 

As well as, we have $x_1 + x_2 + x_3 = 1$

Solving the above equation, we get $x_1 = 0, x_2 =\frac{1}{2}, x_3 = \frac{1}{2}$

 

We know that, in a Markov chain, $A^\infty$ is nothing but all rows steady-state vector.

Therefore, $A^\infty = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} \\\\ 0 & \frac{1}{2} & \frac{1}{2} \\ \\ 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $

 

Ref: If you want to know more about Markov chains and stochastic matrices: https://en.wikipedia.org/wiki/Stochastic_matrix#Definition_and_properties

Comments

@severustux , i don't able to understand your answer, it seems complex to understand. Can you please explain in simple way.

Answer 2

It's  answer should be A i.e ZERO MATRIX,  By simple observation,  values are in fraction will lead to 0.00 something after multiplication, Hence keep multiplying them to infinity(some large value) will lead to ZERO MATRIX. 

Answer 3

$\begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$.$\begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ [calculating $A^{2}$]

=$\begin{bmatrix} \frac{1}{4} &\frac{5}{8} & \frac{1}{8}\\ 0& \frac{5}{8} & \frac{3}{8}\\ 0& \frac{3}{8} & \frac{5}{8} \end{bmatrix}$

=$(\frac{1}{8})^{3}\begin{bmatrix} 2&5 &1 \\ 0&5 &3 \\ 0& 3 & 5 \end{bmatrix}$.$\begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ [calculating $A^{3}$]

=$(\frac{1}{8})^{3}\begin{bmatrix} 1&5 &2 \\ 0&\frac{9}{2} &\frac{7}{2} \\ 0& \frac{7}{2} & \frac{9}{2} \end{bmatrix}$

=$\left ( \frac{1}{8} \right )^{3}\left ( \frac{1}{2} \right )^{2}\begin{bmatrix} 1&5 &2 \\ 0&9 &7 \\ 0& 7 &9 \end{bmatrix}$.$\begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ [calculating $A^{4}$]

=$\left ( \frac{1}{8} \right )^{3}\left ( \frac{1}{2} \right )^{2}\begin{bmatrix} \frac{1}{2}&\frac{17}{4} &\frac{11}{4} \\ 0&\frac{34}{4} &\frac{30}{4} \\ 0& \frac{30}{4} &\frac{34}{4} \end{bmatrix}$

So, we can see here limit exists for matrix ,as value is not tends to infinity

but values of matrix are not all 0s or all 1s

So, ans will be $E)$

Comments

@srestha  mam check my answer !

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@Magma

chk this https://math.stackexchange.com/questions/3238042/limit-of-powers-of-3-times3-matrix/3239473#3239473

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This is wrong answer as D is the answer. Here is a wolfram alpha calculation https://www.wolframalpha.com/input/?i=%7B%7B0.5%2C+0.5%2C0%7D%2C+%7B0%2C+0.75%2C0.25%7D+%2C%7B0%2C+.25%2C+.75%7D%7D%5E100

 

 

Answer 4

Option A

Comments

@Magma

ok

B),C),D) sort out for sure

now, u have some assumption $\lim_{n\rightarrow \infty }\left ( \frac{1}{2} \right )^{n}$ tends to $0$ then A) is true

otherwise E) right?

I truely confused what will they consider

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$0.5^{\infty } = 0$

yeah A or E

but we have generalized the matrix in terms of  'n' but in your case you ain't generalized wrt to 'n'

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it should be done with Markov chains :(