A trust invested some money in two types of bonds. The first bond pays 10% interest and the second bond pays 12%. The trust received Rs 2800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust.

Given that some amount is invested into two types of bonds with 10% and 12% interest rates.

Let the amount invested in bonds of the first type and the second type be Rs x and Rs y respectively.

Hence, the amount invested in each type of the bonds can be represented in matrix form with each column corresponding to a different type of bond as -

The annual interest obtained is Rs 2800.

We know the formula to calculate the interest on a principal of Rs P at a rate R% per annum for t years is given by,

Here, the time is one year and thus T = 1.

Hence, the interest obtained after one year can be expressed in matrix representation as -

⇒ 10x + 12y = 2800 × 100

⇒ 10x + 12y = 280000

∴ 5x + 6y = 140000 …… (1)

However, on reversing the invested amounts, the interest received is Rs 100 less than the earlier value (Rs 2800).

Now, the amount invested in the second bond is Rs x and that in the first bond is Rs y with the annual interest obtained being Rs 2700.

Hence, the interest obtained by exchanging the invested amount of the two bonds after one year can be expressed in matrix representation as –

⇒ 10y + 12x = 2700 × 100

⇒ 12x + 10y = 270000

∴ 6x + 5y = 135000 …… (2)

Recall that the solution to the system of equations that can be written in the form AX = B is given by X = A^–1B.

Here,

We know the inverse of a matrix is given by

|A| = (5)(5) – (6)(6) = 25 – 36 = –11

We have X = A^–1B.

Amount invested in the first bond = x = Rs 10000

Amount invested in the second bond = y = Rs 15000

Thus, the trust invested Rs 10000 in the first bond and Rs 15000 in the second bond.