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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