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	.2	.5	.2	.1
db	.4	.4	.1	.1
sa	.1	.1	.5	.3
x	.3	.3	.1	.3

Assuming that P represents the consumers buying habits over a long period of time, use this transition matrix and the initial-state matrix S0= [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? . . . . .

Let S1 represent the state matrix one week after introducing brand X to the market. Write the equation in terms of P and S0 that can be used to calculate s1

Find the stationary matrix.

S= In the long-run, the total market share for Acme's two soaps is expected to be (%)