Question

A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and $8\frac{1}{2}$% respectively. The total annual interest from these three accounts is ₹550. Equal amounts have been deposited in the 5% and 8% saving accounts. Find the amount deposited in each of the three accounts, with the help of matrices.

Answer 1

​​​Let the amount deposited in each of the three accounts be ₹x, ₹x and ₹y respectively.

Since, the total amount deposited is ₹7,000.
∴ x + x + y = 7000
⇒ 2x + y = 7000 ....(1)

Total annual Interest is ₹550.
∴ $\frac{5}{100}x+\frac{8}{100}x+\frac{17}{200}y=550$
$\Rightarrow 26x+17y=110000$ ....(2)

The above system of equations can be written in matrix form AX = B as
$$\begin{gathered} \left[\begin{array}{cc}2& 1\\ 26& 17\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}7000\\ 110000\end{array}\right] \\ \mathrm{where},A=\left[\begin{array}{cc}2& 1\\ 26& 17\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right]\mathrm{and}B=\left[\begin{array}{c}7000\\ 110000\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{cc}2& 1\\ 26& 17\end{array}\right| \\ =34-26 \\ =8 \\ \mathrm{Let}{C}_{ij}\mathrm{be}\mathrm{the}\mathrm{cofactors}\mathrm{of}\mathrm{elements}{a}_{ij}\mathrm{in}A=\left[a_{ij}\right].\mathrm{Then}, \\ {C}_{11}={\left(-1\right)}^{1+1}17=17,C_{12}={\left(-1\right)}^{1+2}26=-26 \\ {C}_{21}={\left(-1\right)}^{2+1}1=-1,C_{22}={\left(-1\right)}^{2+2}2=2 \\ \mathrm{adj}A={\left[\begin{array}{cc}17& -26\\ -1& 2\end{array}\right]}^T \\ =\left[\begin{array}{cc}17& -1\\ -26& 2\end{array}\right] \\ \Rightarrow A^{-1}=\frac{1}{\left|A\right|}\mathrm{adj}A \\ =\frac{1}{8}\left[\begin{array}{cc}17& -1\\ -26& 2\end{array}\right] \\ X=A^{-1}B \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{cc}17& -1\\ -26& 2\end{array}\right]\left[\begin{array}{c}7000\\ 110000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{c}119000-110000\\ -182000+220000\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{c}9000\\ 38000\end{array}\right] \\ \Rightarrow x=\frac{9000}{8}\mathrm{and}y=\frac{38000}{8} \\ \therefore x=1125\mathrm{and}y=4750. \end{gathered}$$

Hence, the amount deposited in each of the three accounts is ₹1125, ₹1125 and ₹4750.