Acme Soap Company markets one brand of soap, called Standard Acme (SA), and Best Soap Company markets two brands, Standard Best (SB) and Deluxe Best (DB). Currently, Acme has 40% of the market, and the remainder is divided equally between the two Best brands. A proposed new Acme brand, called brand X, was test-marketed in several large cities, producing the transition matrix below for the consumers' weekly buying habits. SB DB SA X SB 0.2 0.3 0.4 0.1 DB 0.3 0.5 0.1 0.1 P = SA 0.2 0.3 0.3 0.2 X 0.5 0.1 0.1 0.3 Assuming that P represents the consumers buying habits over a long period of time, use this transition matrix and the initial-state matrix So = [0.3 0.3 0.4 0 ] to compute successive state matrices in order to approximate the elements in the stationary matrix. If Acme decides to market this new soap, what is the long-run expected total market share for their two soaps? C Let S₁ represent the state matrix one week after introducing brand X to the market. Write the equation in terms of P and So that can be used to calculate S₁. Find the stationary matrix. S= (Type an integer or decimal for each matrix element. Round to two decimal places as needed.) In the long-run, the total market share for Acme's two soaps is expected to be %. (Round to the nearest percent as needed.)

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