A total amount of ₹7000 is deposited in three different saving bank accounts

Question:

A total amount of ₹7000 is deposited in three different saving bank accounts with annual interest rates 5%, 8% and 812">812812% 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.

Solution:

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

Since, the total amount deposited is ₹7,000.

$\therefore x+x+y=7000$

 

$\Rightarrow 2 x+y=7000$         ...(1)

Total annual Interest is ₹550.

$\therefore \frac{5}{100} x+\frac{8}{100} x+\frac{17}{200} y=550$

$\Rightarrow 26 x+17 y=110000$         ....(2)

The above system of equations can be written in matrix form AX = B as

$\left[\begin{array}{cc}2 & 1 \\ 26 & 17\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}7000 \\ 110000\end{array}\right]$

where, $A=\left[\begin{array}{cc}2 & 1 \\ 26 & 17\end{array}\right], X=\left[\begin{array}{l}x \\ y\end{array}\right]$ and $B=\left[\begin{array}{c}7000 \\ 110000\end{array}\right]$

Now,

$|A|=\left|\begin{array}{cc}2 & 1 \\ 26 & 17\end{array}\right|$

$=34-26$

 

$=8$

Let $C_{i j}$ be the cofactors of elements $a_{i j}$ in $A=\left[a_{i j}\right] .$ Then,

$C_{11}=(-1)^{1+1} 17=17, \quad C_{12}=(-1)^{1+2} 26=-26$

$C_{21}=(-1)^{2+1} 1=-1, \quad C_{22}=(-1)^{2+2} 2=2$

$\operatorname{adj} A=\left[\begin{array}{cc}17 & -26 \\ -1 & 2\end{array}\right]^{T}$

$=\left[\begin{array}{cc}17 & -1 \\ -26 & 2\end{array}\right]$

$\Rightarrow A^{-1}=\frac{1}{|A|}$ adj $A$

$=\frac{1}{8}\left[\begin{array}{cc}17 & -1 \\ -26 & 2\end{array}\right]$

$X=A^{-1} B$

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{cc}17 & -1 \\ -26 & 2\end{array}\right]\left[\begin{array}{c}7000 \\ 110000\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{c}119000-110000 \\ -182000+220000\end{array}\right]$

 

$\Rightarrow\left[\begin{array}{l}x \\ y\end{array}\right]=\frac{1}{8}\left[\begin{array}{c}9000 \\ 38000\end{array}\right]$

$\Rightarrow x=\frac{9000}{8}$ and $y=\frac{38000}{8}$

$\therefore x=1125$ and $y=4750$

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

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