Question

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

Answer 1

Let Rs x be invested in the first bond and Rs y be invested in the second bond.
Let A be the investment matrix and B be the interest per rupee matrix. Then,

$$\begin{gathered} \mathrm{A}=\left[\begin{array}{cc}x& y\end{array}\right]\mathrm{and}\mathrm{B}=\left[\begin{array}{c}\frac{10}{100}\\ \frac{12}{100}\end{array}\right] \\ \mathrm{Total}\mathrm{annual}\mathrm{interest}=\mathrm{AB}=\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{c}\frac{10}{100}\\ \frac{12}{100}\end{array}\right]=\frac{10x}{100}+\frac{12y}{100} \\ \therefore \frac{10x}{100}+\frac{12y}{100}=2800 \\ \Rightarrow 10x+12y=280000.....\left(1\right) \end{gathered}$$
If the rates of interest had been interchanged, then the total interest earned is Rs 100 less than the previous interest.
$$\begin{gathered} \therefore \frac{12x}{100}+\frac{10y}{100}=2700 \\ \Rightarrow 12x+10y=270000.....\left(2\right) \end{gathered}$$
The system of equations (1) and (2) can be expressed as
PX = Q, where $P=\left[\begin{array}{cc}10& 12\\ 12& 10\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],Q=\left[\begin{array}{c}280000\\ 270000\end{array}\right]$
$\left|P\right|=\left|\begin{array}{cc}10& 12\\ 12& 10\end{array}\right|=100-144=-44\ne 0$
Thus, P is invertible.
$$\begin{gathered} \therefore X=P^{-1}Q \\ \Rightarrow X=\frac{adjP}{\left|P\right|}Q \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{\left(-44\right)}{\left[\begin{array}{cc}10& -12\\ -12& 10\end{array}\right]}^T\left[\begin{array}{c}280000\\ 270000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{\left(-44\right)}\left[\begin{array}{cc}10& -12\\ -12& 10\end{array}\right]\left[\begin{array}{c}280000\\ 270000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}\frac{2800000-3240000}{-44}\\ \frac{-3360000+2700000}{-44}\end{array}\right]=\left[\begin{array}{c}10000\\ 15000\end{array}\right] \\ \Rightarrow x=10000\mathrm{and}y=15000 \end{gathered}$$

Therefore, Rs 10,000 be invested in the first bond and Rs 15,000 be invested in the second bond.