question_answer

A total amount of Rs. 7000 is deposited in three different saving bank accounts with annual interest rates of 5%, 8% and \[8\frac{1}{2}%,\] respectively. The total annual interest from these three accounts is Rs. 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 matrix multiplication. Keeping nation's growth in mind, justify the value of saving in individual life.

Answer:

Let the amount deposited in the saving bank account with interest rate 5% = x, the amount deposited in the saving bank account with interest rate 8% = y and the amount deposited in the saving bank account with interest rate\[\frac{17}{2}%=z\]. According to the question, \[x+y+z=7000\] \[\Rightarrow \]   \[\frac{5}{100}x+\frac{8}{100}y+\frac{17}{200}=550\] \[\Rightarrow \]   \[x=y\Rightarrow x-y=0\] These equations can be expressed as \[AX=B\] Where \[\therefore \]      \[|A|\,\,=1\left( \frac{17}{200} \right)-1\left( -\frac{17}{200} \right)+1\left( \frac{-\,13}{100} \right)\] \[=\frac{17+17-26}{200}=\frac{8}{200}\ne 0\] So, A is non-singular matrix and inverse exists. \[\therefore \]      \[{{A}_{11}}={{(-\,1)}^{1+1}}\left| \begin{matrix}    8/100 & 17/200  \\    -\,1 & 0  \\ \end{matrix} \right|=\frac{17}{200},\] \[{{A}_{12}}={{(-\,1)}^{1+2}}\left| \begin{matrix}    \frac{5}{100} & \frac{17}{200}  \\    1 & 0  \\ \end{matrix} \right|=\frac{17}{200},\] \[{{A}_{13}}={{(-\,1)}^{1+3}}\left| \begin{matrix}    \frac{5}{100} & \frac{8}{100}  \\    1 & -\,1  \\ \end{matrix} \right|=\frac{-\,13}{100},\] \[{{A}_{21}}={{(-\,1)}^{2+1}}\left| \begin{matrix}    1 & 1  \\    -\,1 & 0  \\ \end{matrix} \right|=-\,1,\] \[{{A}_{22}}={{(-\,1)}^{2+2}}\left| \begin{matrix}    1 & 1  \\    1 & 0  \\ \end{matrix} \right|=-\,1,\] \[{{A}_{23}}={{(-\,1)}^{2+3}}\left| \begin{matrix}    1 & 1  \\    1 & -\,1  \\ \end{matrix} \right|=2,\] \[{{A}_{31}}={{(-\,1)}^{3+1}}\left| \begin{matrix}    1 & 1  \\    \frac{8}{100} & \frac{17}{200}  \\ \end{matrix} \right|=\frac{1}{200},\] \[{{A}_{32}}={{(-\,1)}^{3+2}}\left| \begin{matrix}    1 & 1  \\    \frac{5}{100} & \frac{17}{200}  \\ \end{matrix} \right|=\frac{-\,7}{200},\] \[{{A}_{33}}={{(-\,1)}^{3+3}}\left| \begin{matrix}    1 & 1  \\    \frac{5}{100} & \frac{8}{100}  \\ \end{matrix} \right|=\frac{3}{100}\] Now, adj \[\therefore\] \[\therefore\] \[\therefore \]Amount deposited in the saving bank with interest rate 5% = Rs. 1125 Amount deposited in the saving bank with interest rate 8% = Rs. 1125 Amount deposited in the saving bank with interest rate \[\frac{17}{2}%=Rs.\,\,4750\] Value Habit of saving makes individual self-dependent and nation uses it for betterment of the country as well as its people.