Question

Mr. X has invested a part of his investment in 10% bond A and a part in 15% bond B. His interest income during first year is Rs. 4000. If he invests 20% more in 10% bond A and 10% more in 15% bond B, his income during the second year increases by Rs. 500. Find his initial investment using matrix method

Answer 1

Here, let x and y be the investments in 10% bonds and 15% bonds respectively.
So, according to the given question,
$$\begin{gathered} 0.10x+0.15y=4000 \\ 2x+3y=80000...\left(1\right) \end{gathered}$$

Now, it is given that if he invests 20% more in 10% bond A and 10% more in 15% bond B, his income during the second year increases by Rs 500.
So,
$$\begin{gathered} 0.10\left(x+\frac{x}{5}\right)+0.15\left(y+\frac{y}{10}\right)=4500 \\ 0.10\left(\frac{6x}{5}\right)+0.15\left(\frac{11y}{10}\right)=4500 \end{gathered}$$

Solving further,
$$\begin{gathered} 24x+33y=900000 \\ 3\left(8x+11y\right)=900000 \\ 8x+11y=300000...\left(2\right) \end{gathered}$$

Now, solve equation (1) and (2) using matrices.
$\left[\begin{array}{cc}2& 3\\ 8& 11\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}80000\\ 300000\end{array}\right]$

So,
$$\begin{gathered} X=A^{-1}B \\ X=\left(\frac{1}{\left|A\right|}adjA\right)B \end{gathered}$$

Now,
$$\begin{gathered} \left|A\right|=2\left(11\right)-\left(8\right)\left(3\right) \\ \left|A\right|=22-24 \\ \left|A\right|=-2 \end{gathered}$$

Also,
$adjA=\left[\begin{array}{cc}11& -3\\ -8& 2\end{array}\right]$

Thus,
$$\begin{gathered} X=-\frac{1}{2}\left[\begin{array}{cc}11& -3\\ -8& 2\end{array}\right]\left[\begin{array}{c}80000\\ 300000\end{array}\right] \\ X=-\frac{1}{2}\left[\begin{array}{cc}-\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 4& -1\end{array}\right]\left[\begin{array}{c}80000\\ 300000\end{array}\right] \\ X=\left[\begin{array}{c}-440000+450000\\ 320000-300000\end{array}\right] \\ X=\left[\begin{array}{c}10000\\ 20000\end{array}\right] \end{gathered}$$

Therefore, the new investment in 10% bond A is Rs 10000 and 15% bond B will be Rs 20000