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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 Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question ?

Answers (1)

Let the amount interest by the trust in first and second bond be x and y resp.
Interest from first bond $= \frac{10\times x\times 1}{100}= \frac{10x}{100}$
Interest from second bond $= \frac{12\times y\times 1}{100}= \frac{12y}{100}$
Interest received by trust = Rs 2,800
According to the question, $\frac{10x}{100}+\frac{12y}{100}= 2,800$
$\Rightarrow 10x+12y= 2,80,000---(i)$
and $\frac{12x}{100}+\frac{10y}{100}= 2,700$
$\Rightarrow 12x+10y= 2,70,000---(ii)$
This system of equations can be written in matrix form as follows:
$\begin{bmatrix} 10 &12 \\ 12 & 10 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}= \begin{bmatrix} 2,80,000\\ 2,70,000 \end{bmatrix}$
Now, $\left | A \right |= \begin{vmatrix} 10 & 12\\ 12&10 \end{vmatrix}= 100-144= -44\neq 0$
So, A^-1 exists and the solution of the given system of equations is given by
$X= A^{-1}B$
Let Cij be the cofactor of aij in $A= \left [ aij \right ]$ . Then
$c_{11}= 10,c_{12}= -12,c_{22}= 10$
$\therefore adj A= \begin{bmatrix} 10 &-12 \\ -12& 10 \end{bmatrix}^{T}= \begin{bmatrix} 10 &-12 \\ -12&10 \end{bmatrix}$
So, $A^{-1}= \frac{1}{\left | A \right |}\left ( adjA \right )= \frac{-1}{44}\begin{bmatrix} 10 &-12 \\ -12& 10 \end{bmatrix}$
Hence the solution is given by
$X= A^{-1}B=\frac{-1}{44}\begin{bmatrix} 10 &-12 \\ -12& 10 \end{bmatrix}\begin{bmatrix} 2,80,000\\ 2,70,000 \end{bmatrix}$
$\begin{bmatrix} x\\ y \end{bmatrix}= \frac{-1}{44}\begin{bmatrix} 28,00,000 -32,40,000 \\ -33,60,000 +27,00,000 \end{bmatrix}$
$\begin{bmatrix} x\\ y \end{bmatrix}= \frac{-1}{44}\begin{bmatrix} -4,40,000\\ -6,60,600 \end{bmatrix}= \begin{bmatrix} 10,000\\ 15,000 \end{bmatrix}$
$\Rightarrow x= 10,000\, \S \, y= 15,000$
$\Rightarrow A= x+y= 10,000+15,000= Rs\, 25,000$
Hence the amount interest by the trust is Rs 25,000 Value: giving help to those in need is a humanlisation art.