**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.

**Solution:**

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,

$\mathrm{A}=\left[\begin{array}{ll}x & y\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\frac{10}{100} \\ \frac{12}{100}\end{array}\right]$

Total annual interest $=\mathrm{AB}=\left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{c}\frac{10}{100} \\ \frac{12}{100}\end{array}\right]=\frac{10 z}{100}+\frac{12 y}{100}$

$\therefore \frac{10 x}{100}+\frac{12 y}{100}=2800$

$\Rightarrow 10 x+12 y=280000$ ....(1)

If the rates of interest had been interchanged, then the total interest earned is Rs 100 less than the previous interest.

$\therefore \frac{12 x}{100}+\frac{10 y}{100}=2700$

$\Rightarrow 12 x+10 y=270000$ ....(2)

The system of equations (1) and (2) can be expressed as

$P X=Q$, where $P=\left[\begin{array}{ll}10 & 12 \\ 12 & 10\end{array}\right], X=\left[\begin{array}{l}x \\ y\end{array}\right], Q=\left[\begin{array}{l}280000 \\ 270000\end{array}\right]$

$|P|=\left|\begin{array}{ll}10 & 12 \\ 12 & 10\end{array}\right|=100-144=-44 \neq 0$

Thus, $P$ is invertible.

$\therefore X=P^{-1} Q$

$\Rightarrow X=\frac{a d j P}{|P|} Q$

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{(-44)}\left[\begin{array}{cc}10 & -12 \\ -12 & 10\end{array}\right]^{T}\left[\begin{array}{l}280000 \\ 270000\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{(-44)}\left[\begin{array}{cc}10 & -12 \\ -12 & 10\end{array}\right]\left[\begin{array}{l}280000 \\ 270000\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}\frac{2800000-3240000}{-44} \\ \frac{-336000+2700000}{-44}\end{array}\right]=\left[\begin{array}{l}10000 \\ 15000\end{array}\right]$

$\Rightarrow x=10000$ and $y=15000$

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

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