Use an Inverse Matrix to Find the Solution to the System

question 129

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 tl 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 $\mathrm { A } = 81,000$ customers and Company 2 has $\mathrm { B } = 153,000$ customers this year, then how many customers did Company 2 have last year?

• A) 162,000 customers
• B) 76,000 customers
• C) 72,000 customers
• D) 168,000 customers

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Definitions:

Compact Bone

Dense bone tissue that makes up the outer layer of bones; it is smooth, solid, and gives bones their strength.

Cartilage

A flexible, connective tissue found in various parts of the body, including joints, rib cage, and earlobes, providing support and allowing for movement.

Bone Marrow

A soft, spongy tissue inside bones that produces blood cells.

Lamellae

Thin layers or plates of tissue or bone, often referring to the concentric layers found in bone or the layers within the chloroplasts of plant cells.