Question

An amount of rs 5000 is put into three investments at the rate of interests of 6.7% , 7.7% and 8% per annum respectively. The total annual income is rs 350. If the combined income from the first two investments is rs 70 more than the income from the third , find the amount of each investment by matrix method.​

Answer 1

Given:

An amount of rs 5000 is put into three investments at the rate of interests of 6.7%, 7.7% and 8% per annum respectively.

The total annual income is rs 350.

To find:

If the combined income from the first two investments is rs 70 more than the income from the third, find the amount of each investment by matrix method.​

Solution:

Let the investments be x, y and z.

x = 6.7% × 5000 = 6.7/100 × 5000 = 335

y = 7.7% × 5000 = 7.7/100 × 5000 = 385

z = 8% × 5000 = 8/100 × 5000 = 400

6.7x + 7.7y + 8z = 100

x + y + z = 350

x + y - z = 70

AX = B

$\left[\begin{array}{ccc}6.7&7.7&8\\1&1&1\\1&1&-1\end{array}\right] \begin{bmatrix}x\\ y\\ z\end{bmatrix} = \begin{bmatrix}100\\ 350\\ 70\end{bmatrix}$

$\Delta =\begin{bmatrix}6.7&7.7&8\\ 1&1&1\\ 1&1&-1\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}-7.7\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}1&1\\ 1&1\end{pmatrix}\\=6.7\left(-2\right)-7.7\left(-2\right)+8\cdot \:0\\\Delta = 2$

$\Delta_1 =\begin{bmatrix}100&7.7&8\\ 350&1&1\\ 70&1&-1\end{bmatrix}\\=100\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}-7.7\cdot \det \begin{pmatrix}350&1\\ 70&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}350&1\\ 70&1\end{pmatrix}\\=100\left(-2\right)-7.7\left(-420\right)+8\cdot \:280\ \\\Delta_1 = 5274$

$\Delta_2 =\begin{bmatrix}6.7&100&8\\ \:\:1&350&1\\ \:\:1&70&-1\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}350&1\\ 70&-1\end{pmatrix}-100\cdot \det \begin{pmatrix}1&1\\ 1&-1\end{pmatrix}+8\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}\\=6.7\left(-420\right)-100\left(-2\right)+8\left(-280\right)\ \\\Delta_2 = -4854$

$\Delta_3 =\begin{bmatrix}6.7&7.7&100\\ 1&1&350\\ 1&1&70\end{bmatrix}\\=6.7\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}-7.7\cdot \det \begin{pmatrix}1&350\\ 1&70\end{pmatrix}+100\cdot \det \begin{pmatrix}1&1\\ 1&1\end{pmatrix}\\=6.7\left(-280\right)-7.7\left(-280\right)+100\cdot \:0\ \\\Delta_3 = 280$

Now let us consider,

Hence the investments are as follows:

x = Δ1/Δ = 5274/2 = 2637

y = Δ2/Δ = -4854/2 = -2427

z = Δ3/Δ = 280/2 = 140

Verification:

The combined income from the first two investments is rs 70 more than the income from the third.

x + y = z + 70

2637 + (-2427) = 140 + 70

210 = 210