Samacheer Kalvi Class 12 Business Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.3

Question 1. The subscription of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 45% of those who already subscribe will subscribe again while 30% of those who do not subscribe. On the last letter, it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?
Answer: To find the percentage of people expected to subscribe this year, we use a transition probability matrix. Let S represent subscribers and F represent non-subscribers.
The transition matrix \( T \) is set up as follows:
\( S \to S \quad 45\% = 0.45 \)
\( S \to F \quad 100\% - 45\% = 55\% = 0.55 \)
\( F \to S \quad 30\% = 0.30 \)
\( F \to F \quad 100\% - 30\% = 70\% = 0.70 \)
\[ T = \begin{pmatrix} 0.45 & 0.55 \\ 0.30 & 0.70 \end{pmatrix} \]
The initial state vector \( (S \quad F) \) from the last letter is \( (0.40 \quad 0.60) \) because 40% ordered a subscription, meaning 60% did not. So, 40% are subscribers and 60% are non-subscribers.
Now, we calculate the state after one year by multiplying the initial state vector by the transition matrix:
\[ (S \quad F)_{new} = (0.40 \quad 0.60) \begin{pmatrix} 0.45 & 0.55 \\ 0.30 & 0.70 \end{pmatrix} \]
\[ = (0.40 \times 0.45 + 0.60 \times 0.30 \quad 0.40 \times 0.55 + 0.60 \times 0.70) \]
\[ = (0.180 + 0.180 \quad 0.220 + 0.420) \]
\[ = (0.36 \quad 0.64) \]
This means that after one year, 36% of those receiving the current letter can be expected to subscribe, and 64% can be expected not to subscribe. This method is useful for predicting future outcomes based on current trends.
In simple words: We set up a grid that shows how likely people are to subscribe again or to start subscribing. We start with how many people subscribed last time (40%). Then, we do a math step to see how many will subscribe this time. The answer is that 36% of people are expected to subscribe.

🎯 Exam Tip: When dealing with transition matrices, carefully identify the initial state vector and ensure the matrix multiplication order is correct (state vector multiplied by the transition matrix).

Question 2. A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use he transit system. Of those who use metro train this year and 70% will continue to use metro train next year and 30% will switch over to transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train this year.
(i) What percent of commuters will be using the transit system after one year?
(ii) What percent of commuters will be using the transit system in the one run?
Answer: First, let's define the states: S for Transit System and C for Metro Train.
The transition probability matrix \( T \) describes how commuters switch or stay:
From S to S: 70% (0.7)
From S to C: 30% (0.3)
From C to S: 30% (0.3)
From C to C: 70% (0.7)
\[ T = \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix} \]
The current market share (initial state vector) is \( (S \quad C) = (0.6 \quad 0.4) \), as 60% use the transit system and 40% use the metro train.

(i) Percent of commuters using transit system after one year:
To find the shares after one year, we multiply the current state vector by the transition matrix:
\[ (S_1 \quad C_1) = (0.6 \quad 0.4) \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix} \]
\[ = (0.6 \times 0.7 + 0.4 \times 0.3 \quad 0.6 \times 0.3 + 0.4 \times 0.7) \]
\[ = (0.42 + 0.12 \quad 0.18 + 0.28) \]
\[ = (0.54 \quad 0.46) \]
So, after one year, 54% of commuters will be using the transit system, and 46% will be using the metro train. This prediction helps urban planners understand immediate impacts.

(ii) Percent of commuters using transit system in the long run (equilibrium):
At equilibrium, the market shares become stable and do not change. Let the equilibrium state be \( (S \quad C) \), where \( S+C=1 \).
The condition for equilibrium is \( (S \quad C) T = (S \quad C) \).
\[ (S \quad C) \begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix} = (S \quad C) \]
This gives us two equations:
1. \( 0.7S + 0.3C = S \)
2. \( 0.3S + 0.7C = C \)
Using equation 1:
\( 0.3C = S - 0.7S \)
\( \implies 0.3C = 0.3S \)
\( \implies C = S \)
Since \( S+C=1 \), we substitute \( C=S \):
\( S+S = 1 \)
\( \implies 2S = 1 \)
\( \implies S = 0.50 \)
Since \( C=S \), then \( C = 0.50 \).
Therefore, in the long run, 50% of commuters will be using the transit system. Over time, the usage shares will stabilize at these percentages.
In simple words: (i) We use a special math grid to find out how many people will use the transit system next year. Right now, 60% use it. After calculating, we expect 54% to use it next year.
(ii) For a very long time, the number of people using the transit system will settle down. This steady number will be 50% of all commuters.

🎯 Exam Tip: Remember that equilibrium is reached when the state vector remains unchanged after multiplication by the transition matrix, i.e., \( PT = P \). Always ensure \( S+C=1 \) when solving for equilibrium states.

Question 3. Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% contionues to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?
Answer: Let A represent soap A and B represent soap B.
The transition probability matrix \( T \) based on customer switching behavior is:
From A to A: 65% (0.65)
From A to B: 35% (0.35)
From B to A: 45% (0.45)
From B to B: 55% (0.55)
\[ T = \begin{pmatrix} 0.65 & 0.35 \\ 0.45 & 0.55 \end{pmatrix} \]
The present market shares (initial state vector) are \( (A \quad B) = (0.15 \quad 0.85) \).

Market shares after one year:
We multiply the initial state vector by the transition matrix:
\[ (A_1 \quad B_1) = (0.15 \quad 0.85) \begin{pmatrix} 0.65 & 0.35 \\ 0.45 & 0.55 \end{pmatrix} \]
\[ = (0.15 \times 0.65 + 0.85 \times 0.45 \quad 0.15 \times 0.35 + 0.85 \times 0.55) \]
\[ = (0.0975 + 0.3825 \quad 0.0525 + 0.4675) \]
\[ = (0.48 \quad 0.52) \]
So, after one year, the market share for soap A will be 48%, and for soap B will be 52%. This calculation shows the short-term market shift.

Equilibrium market shares (long run):
At equilibrium, the market shares stabilize and no longer change. Let the equilibrium state be \( (A \quad B) \), where \( A+B=1 \).
The condition for equilibrium is \( (A \quad B) T = (A \quad B) \).
\[ (A \quad B) \begin{pmatrix} 0.65 & 0.35 \\ 0.45 & 0.55 \end{pmatrix} = (A \quad B) \]
This gives us the equations:
1. \( 0.65A + 0.45B = A \)
2. \( 0.35A + 0.55B = B \)
From equation 1:
\( 0.45B = A - 0.65A \)
\( \implies 0.45B = 0.35A \)
Since \( A+B=1 \), we know \( B = 1-A \). Substitute this into the equation:
\( 0.45(1-A) = 0.35A \)
\( \implies 0.45 - 0.45A = 0.35A \)
\( \implies 0.45 = 0.35A + 0.45A \)
\( \implies 0.45 = 0.80A \)
\( \implies A = \frac{0.45}{0.80} = \frac{45}{80} = \frac{9}{16} \)
\( \implies A = 0.5625 \), or 56.25%.
Then \( B = 1-A = 1 - 0.5625 = 0.4375 \), or 43.75%.
Therefore, at equilibrium, the market share for soap A will be 56.25%, and for soap B will be 43.75%. This represents the stable market distribution if current trends continue indefinitely.
In simple words: After one year, soap A will have 48% of the market, and soap B will have 52%. If these buying patterns continue, over a long time, soap A will settle at 56.25% of the market, and soap B will settle at 43.75%.

🎯 Exam Tip: When calculating market shares after a specific period, perform matrix multiplication. For equilibrium, set up the equation \( (A \quad B) T = (A \quad B) \) and use \( A+B=1 \) to solve for the stable percentages.

Question 4. Two products A and B currently share the market with shares of 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again whereas 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?
Answer: Let A represent product A and B represent product B.
The transition probability matrix \( T \) based on weekly brand switching is:
From A to A: 60% (0.60)
From A to B: 40% (0.40)
From B to A: 20% (0.20)
From B to B: 80% (0.80)
\[ T = \begin{pmatrix} 0.60 & 0.40 \\ 0.20 & 0.80 \end{pmatrix} \]
The initial market shares are \( (A_0 \quad B_0) = (0.50 \quad 0.50) \).

Shares after one week:
Multiply the initial state by the transition matrix:
\[ (A_1 \quad B_1) = (0.50 \quad 0.50) \begin{pmatrix} 0.60 & 0.40 \\ 0.20 & 0.80 \end{pmatrix} \]
\[ = (0.50 \times 0.60 + 0.50 \times 0.20 \quad 0.50 \times 0.40 + 0.50 \times 0.80) \]
\[ = (0.30 + 0.10 \quad 0.20 + 0.40) \]
\[ = (0.40 \quad 0.60) \]
After one week, the shares are 40% for A and 60% for B.

Shares after two weeks:
Now, we use the shares after one week as the new initial state and multiply again by the transition matrix:
\[ (A_2 \quad B_2) = (0.40 \quad 0.60) \begin{pmatrix} 0.60 & 0.40 \\ 0.20 & 0.80 \end{pmatrix} \]
\[ = (0.40 \times 0.60 + 0.60 \times 0.20 \quad 0.40 \times 0.40 + 0.60 \times 0.80) \]
\[ = (0.24 + 0.12 \quad 0.16 + 0.48) \]
\[ = (0.36 \quad 0.64) \]
After two weeks, the shares are 36% for A and 64% for B. These calculations show the short-term evolution of market shares.

Equilibrium (long run):
At equilibrium, the market shares become stable. Let the equilibrium state be \( (A \quad B) \), where \( A+B=1 \).
The condition for equilibrium is \( (A \quad B) T = (A \quad B) \).
\[ (A \quad B) \begin{pmatrix} 0.60 & 0.40 \\ 0.20 & 0.80 \end{pmatrix} = (A \quad B) \]
This gives us the equations:
1. \( 0.60A + 0.20B = A \)
2. \( 0.40A + 0.80B = B \)
From equation 1:
\( 0.20B = A - 0.60A \)
\( \implies 0.20B = 0.40A \)
\( \implies B = \frac{0.40}{0.20}A \)
\( \implies B = 2A \)
Since \( A+B=1 \), we substitute \( B=2A \):
\( A + 2A = 1 \)
\( \implies 3A = 1 \)
\( \implies A = \frac{1}{3} \approx 0.3333 \)
So, \( A \approx 33.33\% \).
Then \( B = 1-A = 1 - \frac{1}{3} = \frac{2}{3} \approx 0.6667 \).
So, \( B \approx 66.67\% \).
Therefore, in the long run, product A will have approximately 33.33% of the market, and product B will have approximately 66.67%. This indicates the stable market distribution if the switching patterns continue.
In simple words: Both products A and B start with 50% of the market. After one week, product A will have 40% and B will have 60%. After two weeks, A will have 36% and B will have 64%. If these patterns keep going for a very long time, product A will end up with about 33.33% of the market, and product B will have about 66.67%.

🎯 Exam Tip: For questions asking for shares after multiple periods, repeatedly apply matrix multiplication, using the result of the previous period as the new initial state. For equilibrium, use the \( PT=P \) and \( A+B=1 \) approach.