Exercise $5\mathrm{.}9$

A financial institution offers three types of investment products: Savings Account, Fixed Deposit, and Mutual Fund. The institution has determined the interest rates and investment limits for each product as follows:

Savings Account: Interest rate $=2\%$ per annum, maximum investment $=$$$10,000$

Fixed Deposit: Interest rate $=5\%$ per annum, maximum investment $=$$$50,000$

Mutual Fund: Interest rate $=8\%$ per annum, maximum investment $=$$$100,000$The institution aims to maximize its interest income while adhering to the following constraints:

The total investment in Savings Accounts should not exceed $$30,000.$

The total investment in Fixed Deposits should not exceed $$80,000.$

The total investment in Mutual Funds should not exceed $$150,000.$

Let $x,y,$ and $z$ represent the amounts invested in Savings Account, Fixed Deposit, and Mutual Fund respectively.

$($a$)$ Formulate a system of linear equations representing the investment constraints described above.

$($b$)$ Represent the system of equations in matrix form using coefficient matrices.

$($c$)$ Solve the system of equations using matrix algebra to determine the optimal investment amounts for Savings Account, Fixed Deposit, and Mutual Fund.

$($d$)$ Interpret your results in the context of the financial institution's investment strategy.