A credit union classifies automobile loans into one of four categories: the loan has been paid in full (F), the account is in good standing (G) with all payments up to date, the account is in arrears (A) with one or more missing payments, or the account has been classified as a bad debt (B) and sold to a collection agency. Past records indicate that each month 10% of the accounts in good standing pay the loan in full, 80% remain in good standing, and 10% become in arrears. Furthermore, 10% of the accounts in arrears are paid in full, 40% become accounts in good standing, 40% remain in arrears, and 10$ are classified as bad debts.

a) give the transition matrix for the described Markov chain.
b) what is the probability that an account currently in arrears will eventually be paid in full?
c) what is the probability that an account in good standing will end up becoming a bad debt hat is sold to a collection agency?


for a. the transition matrix i got was:
T =
[ .4 .1 .1 .4 ]
| 0 1 0 0 |
| 0 0 1 0 |
[ .1 0 .1 .8 ]

for b. the final matrix i got was
[ 1/3 5/6 ]
[ 1/4 1 ]

this is where i got stuck. the rows for part b are supposed to equal 1 and they dont. i tried all my calculations again and am still stuck. help?