Use an Inverse Matrix to Find the Solution to the System

question 13

Multiple Choice

Use an inverse matrix to find the solution to the system.
-Two phone companies compete for customers in a city. Company 1 retains $3 / 4$ of its customers and loses $1 / 4$ of its customers to Company 2 . Company 2 retains $5 / 6$ of its customers and loses $1 / 6$ to Company 1 . If we represent the fraction of the market held last year by $\left[ \begin{array} { l } a \\ b \end{array} \right]$ , where $a$ is the number the that Company 1 had last year and $b$ is number that Company 2 had last year, then the number that each company will have this year can be found by t] following matrix equation.
$$\left[ \begin{array} { l } \mathrm { A } \\\mathrm { B }\end{array} \right] = \left[ \begin{array} { l l } 3 / 4 & 1 / 6 \\1 / 4 & 5 / 6\end{array} \right] \left[ \begin{array} { l } \mathrm { a } \\\mathrm { b }\end{array} \right]$$
If Company 1 has $\mathrm { A } = 90,000$ customers and Company 2 has $\mathrm { B } = 198,000$ customers this year, how many customers did Company 1 have last year?

• A) 76,000 customers
• B) 222,000 customers
• C) 72,000 customers
• D) 216,000 customers

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Definitions:

Reinforcement

In the context of behavior, it is the process of encouraging or establishing a pattern of behavior through rewards or consequences.

Stimulus Discrimination

The ability of an individual to differentiate between similar stimuli and respond appropriately to each, often used in marketing to distinguish a brand from its competitors.

Cognitive Dissonance

The mental discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values at the same time, leading to alteration in one of the attitudes, beliefs, or behaviors to reduce the discomfort.

Selective Retention

The tendency of individuals to remember more information that is personally meaningful to them while forgetting irrelevant information.