Math 307 – Sec. 202 (TZOU)FINAL — April 18, 2017Page 2 of 151. In a warm climate where it never snows, the weather on any particular day is eithersunny, cloudy, or rainy (assume that the weather is constant for the entire day). If it issunny, there is a 1/2 chance that it will remain sunny the next day, 1/3 chance that itwill be cloudy the next day, and 1/6 chance that it will rain the next day. If it is cloudy,the next day will be sunny, cloudy, or rainy with equal probability. If it is raining, thereis a 1/6 chance that it is sunny the next day, a 1/3 chance that it is cloudy the nextday, and a 1/2 chance that it remains rainy the next day. Denote the probability of sun,cloud, and rain on thek-th day bysk,ck, andrk, respectively. Denote also the vectorvk= (sk, ck, rk)T.(a)2 marksShow thatλ= 1 must be an eigenvalue ofanyn×nmatrixMwhose columns eachsum to one. That is, show that if∑ni=1Mij= 1 forj= 1, . . . , n, thenλ= 1 is aneigenvalue ofM.(b)3 marksConstruct the transition matrixAsuch thatvk+1=Avk(you will receive 1 pointfor each correct column). Here,vkis defined above.(c)2 marksThe eigenpairs ofAare such thatAx1=λ1x1,Ax2=λ2x2, andAx3=λ3x3, whereλ1= 1, and|λ2|,|λ3|<1.Suppose the eigenvectorsx1,x2, andx3are mutuallyorthogonal (but not necessarily orthonormal). For arbitrary initial distributionv0,find the coefficientsd1, d2, andd3such thatv0=d1x1+d2x2+d3x3.Do notleave your answer in terms of the inverse of a certain matrix.Do not compute theeigenvectors.(d)2 marksUse the result of Part (c) to expressv20in terms ofx1,x2, andx3, along with thecoefficientsd1,d2, andd3.Do not compute the eigenvectors.(e)1 markExpress the steady state distributionv∞in terms ofx1and/orx2and/orx3, alongwith the coefficientsd1and/ord2and/ord3.Do not compute the eigenvectors.