A credit union classifies car 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$$ payments, or the account has been classified as a bad debt$($B$)$ and sold to a collection agency. Past records indicated that each$$ month, $20\%$ of the accounts in good standing pay the loan in$$ full, $70\%$ remain in good$$ standing, and $10\%$ become in arrears.$$ Furthermore, $20\%$ of the accounts in arrears are paid in$$ full, $30\%$ become accounts in good$$ standing, $40\%$ remain in$$ arrears, and $10\%$ are classified as bad debts. Complete parts$($A$)$ through$($C$)$ below.

$($A$)$ In the long$$ run, what percentage of the accounts in arrears will pay their loan in$$ full?

Present the given information in a standard form matrix. Within the group of absorbing states and within the group of nonabsorbing$$ states, from left to right and top to$$ bottom, list F before$$ A$,$ B$,$ and$$ G; list B before A and$$ G; and list G before A$.$

Now determine the limiting matrix Upper P overbar.

In the long$$ run, what $\%$ of the accounts in arrears will pay their loan in full.

Part $4$

$($B$)$ In the long$$ run, what percentage of the accounts in good standing will become bad$$ debts?

Part $5$

$($C$)$ What is the average number of months that an account in arrears will remain in this system before it is either paid in full or classified as a bad$$ debt?