Question #213104

Question 1


Stonewall receives ¢250 per year in simple interest from an amount of money he invested in ADB, Barclays and GCB. Suppose ADB pays an interest of 2%, Barclays pays an interest of 4% and GCB pays an interest of 5% per annum and an amount of ¢350 more was invested in Barclays than the amount invested in ADB and GCB combined. Also, the amount invested in Barclays is 2 times the amount invested in GCB.


a) Write down the three linear equations and represent them in the matrix form ?? = ?.


b) Find the amount of money Stonewall invested in ADB, Barclays and GCB using Matrix inversion.






1
Expert's answer
2021-07-06T03:24:12-0400

Lets x - invested in ADB,

y - invested inBarclays ,

z - invested in GCB.

0.02x+0.04y+0.05z=2500.02x+0.04y+0.05z=250

y=350+(x+z)y=350+(x+z)

y=2zy=2z

We have a system

0.02x+0.04y+0.05z=250x+yz=350y2z=00.02x+0.04y+0.05z=250\\ -x+y-z=350\\ y-2z=0

A=(0.020.040.05111012),X=(xyz),B=(2503500)A=\begin{pmatrix} 0.02 & 0.04&0.05 \\ -1& 1&-1\\ 0&1&-2 \end{pmatrix}, X=\begin{pmatrix} x \\ y\\ z \end{pmatrix}, \\B=\begin{pmatrix} 250\\ 350\\ 0 \end{pmatrix}

AX=BX=A1BA1:AX=B\\ X=A^{-1}\cdot B\\ A^{-1}:


(0.020.040.05100111010012001)\begin{pmatrix} 0.02 & 0.04&0.05&|1&0&0 \\ -1& 1&-1&|0&1&0\\ 0&1&-2&|0&0&1 \end{pmatrix}\leftrightarrow

Ir\leftrightarrowIIr

(1110100.020.040.05100012001)\begin{pmatrix} -1&1&-1&|0&1&0 \\ 0.02&0.04&0.05&|1&0&0\\ 0&1&-2&|0&0&1 \end{pmatrix}\leftrightarrow

IIr+Ir(0.02)

Ir(-1)

(11101000.060.0310.020012001)\begin{pmatrix} 1&-1&1&|0&-1&0 \\ 0&0.06&0.03&|1&0.02&0\\ 0&1&-2&|0&0&1 \end{pmatrix}\leftrightarrow

IIr\leftrightarrow IIIr

(11101001200100.060.0310.020)\begin{pmatrix} 1&-1&1&|0&-1&0 \\ 0&1&-2&|0&0&1\\ 0&0.06&0.03&|1&0.02&0\\ \end{pmatrix}\leftrightarrow

IIIr+IIr(-0.06)

(111010012001000.1510.020.06)\begin{pmatrix} 1&-1&1&|0&-1&0 \\ 0&1&-2&|0&0&1\\ 0&0&0.15&|1&0.02&-0.06\\ \end{pmatrix}\leftrightarrow

IIIr(203\frac{20}{3})

(11101001200100120321525)\begin{pmatrix} 1&-1&1&|0&-1&0 \\ 0&1&-2&|0&0&1\\ 0&0&1&|\frac{20}{3}&\frac{2}{15}&-\frac{2}{5}\\ \end{pmatrix}\leftrightarrow

IIr+IIIr(2)

Ir+IIIr(-1)

(1102031715250104034151500120321525)\begin{pmatrix} 1&-1&0&|-\frac{20}{3}&-\frac{17}{15}&\frac{2}{5} \\ 0&1&0&|\frac{40}{3}&\frac{4}{15}&\frac{1}{5}\\ 0&0&1&|\frac{20}{3}&\frac{2}{15}&-\frac{2}{5}\\ \end{pmatrix}\leftrightarrow

Ir+IIr

(1002031315350104034151500120321525)\begin{pmatrix} 1&0&0&|\frac{20}{3}&-\frac{13}{15}&\frac{3}{5} \\ 0&1&0&|\frac{40}{3}&\frac{4}{15}&\frac{1}{5}\\ 0&0&1&|\frac{20}{3}&\frac{2}{15}&-\frac{2}{5}\\ \end{pmatrix}


A1=(2031315354034151520321525)A^{-1}=\begin{pmatrix} \frac{20}{3}&-\frac{13}{15}&\frac{3}{5} \\ \frac{40}{3}&\frac{4}{15}&\frac{1}{5}\\ \frac{20}{3}&\frac{2}{15}&-\frac{2}{5}\\ \end{pmatrix}

X=(2031315354034151520321525)(2503500)==(4090310280351403)X=\begin{pmatrix} \frac{20}{3}&-\frac{13}{15}&\frac{3}{5} \\ \frac{40}{3}&\frac{4}{15}&\frac{1}{5}\\ \frac{20}{3}&\frac{2}{15}&-\frac{2}{5}\\ \end{pmatrix}\cdot\begin{pmatrix} 250\\ 350\\ 0 \end{pmatrix}=\\ =\begin{pmatrix} \frac{4090}{3}\\ \frac{10280}{3}\\ \frac{5140}{3} \end{pmatrix}

x=136313\frac{1}{3}

y=342623\frac{2}{3}

z=171313\frac{1}{3}



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