Lets x - invested in ADB,
y - invested inBarclays ,
z - invested in GCB.
0.02 x + 0.04 y + 0.05 z = 250 0.02x+0.04y+0.05z=250 0.02 x + 0.04 y + 0.05 z = 250
y = 350 + ( x + z ) y=350+(x+z) y = 350 + ( x + z )
y = 2 z y=2z y = 2 z
We have a system
0.02 x + 0.04 y + 0.05 z = 250 − x + y − z = 350 y − 2 z = 0 0.02x+0.04y+0.05z=250\\
-x+y-z=350\\
y-2z=0 0.02 x + 0.04 y + 0.05 z = 250 − x + y − z = 350 y − 2 z = 0
A = ( 0.02 0.04 0.05 − 1 1 − 1 0 1 − 2 ) , X = ( x y z ) , B = ( 250 350 0 ) 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} A = ⎝ ⎛ 0.02 − 1 0 0.04 1 1 0.05 − 1 − 2 ⎠ ⎞ , X = ⎝ ⎛ x y z ⎠ ⎞ , B = ⎝ ⎛ 250 350 0 ⎠ ⎞
A X = B X = A − 1 ⋅ B A − 1 : AX=B\\
X=A^{-1}\cdot B\\
A^{-1}: A X = B X = A − 1 ⋅ B A − 1 :
( 0.02 0.04 0.05 ∣ 1 0 0 − 1 1 − 1 ∣ 0 1 0 0 1 − 2 ∣ 0 0 1 ) ↔ \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 ⎝ ⎛ 0.02 − 1 0 0.04 1 1 0.05 − 1 − 2 ∣1 ∣0 ∣0 0 1 0 0 0 1 ⎠ ⎞ ↔
Ir↔ \leftrightarrow ↔ IIr
( − 1 1 − 1 ∣ 0 1 0 0.02 0.04 0.05 ∣ 1 0 0 0 1 − 2 ∣ 0 0 1 ) ↔ \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 ⎝ ⎛ − 1 0.02 0 1 0.04 1 − 1 0.05 − 2 ∣0 ∣1 ∣0 1 0 0 0 0 1 ⎠ ⎞ ↔
IIr+Ir(0.02)
Ir(-1)
( 1 − 1 1 ∣ 0 − 1 0 0 0.06 0.03 ∣ 1 0.02 0 0 1 − 2 ∣ 0 0 1 ) ↔ \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 ⎝ ⎛ 1 0 0 − 1 0.06 1 1 0.03 − 2 ∣0 ∣1 ∣0 − 1 0.02 0 0 0 1 ⎠ ⎞ ↔
IIr↔ \leftrightarrow ↔ IIIr
( 1 − 1 1 ∣ 0 − 1 0 0 1 − 2 ∣ 0 0 1 0 0.06 0.03 ∣ 1 0.02 0 ) ↔ \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 ⎝ ⎛ 1 0 0 − 1 1 0.06 1 − 2 0.03 ∣0 ∣0 ∣1 − 1 0 0.02 0 1 0 ⎠ ⎞ ↔
IIIr+IIr(-0.06)
( 1 − 1 1 ∣ 0 − 1 0 0 1 − 2 ∣ 0 0 1 0 0 0.15 ∣ 1 0.02 − 0.06 ) ↔ \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 ⎝ ⎛ 1 0 0 − 1 1 0 1 − 2 0.15 ∣0 ∣0 ∣1 − 1 0 0.02 0 1 − 0.06 ⎠ ⎞ ↔
IIIr(20 3 \frac{20}{3} 3 20 )
( 1 − 1 1 ∣ 0 − 1 0 0 1 − 2 ∣ 0 0 1 0 0 1 ∣ 20 3 2 15 − 2 5 ) ↔ \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 ⎝ ⎛ 1 0 0 − 1 1 0 1 − 2 1 ∣0 ∣0 ∣ 3 20 − 1 0 15 2 0 1 − 5 2 ⎠ ⎞ ↔
IIr+IIIr(2)
Ir+IIIr(-1)
( 1 − 1 0 ∣ − 20 3 − 17 15 2 5 0 1 0 ∣ 40 3 4 15 1 5 0 0 1 ∣ 20 3 2 15 − 2 5 ) ↔ \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 ⎝ ⎛ 1 0 0 − 1 1 0 0 0 1 ∣ − 3 20 ∣ 3 40 ∣ 3 20 − 15 17 15 4 15 2 5 2 5 1 − 5 2 ⎠ ⎞ ↔
Ir+IIr
( 1 0 0 ∣ 20 3 − 13 15 3 5 0 1 0 ∣ 40 3 4 15 1 5 0 0 1 ∣ 20 3 2 15 − 2 5 ) \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} ⎝ ⎛ 1 0 0 0 1 0 0 0 1 ∣ 3 20 ∣ 3 40 ∣ 3 20 − 15 13 15 4 15 2 5 3 5 1 − 5 2 ⎠ ⎞
A − 1 = ( 20 3 − 13 15 3 5 40 3 4 15 1 5 20 3 2 15 − 2 5 ) 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} A − 1 = ⎝ ⎛ 3 20 3 40 3 20 − 15 13 15 4 15 2 5 3 5 1 − 5 2 ⎠ ⎞
X = ( 20 3 − 13 15 3 5 40 3 4 15 1 5 20 3 2 15 − 2 5 ) ⋅ ( 250 350 0 ) = = ( 4090 3 10280 3 5140 3 ) 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 = ⎝ ⎛ 3 20 3 40 3 20 − 15 13 15 4 15 2 5 3 5 1 − 5 2 ⎠ ⎞ ⋅ ⎝ ⎛ 250 350 0 ⎠ ⎞ = = ⎝ ⎛ 3 4090 3 10280 3 5140 ⎠ ⎞
x=13631 3 \frac{1}{3} 3 1
y=34262 3 \frac{2}{3} 3 2
z=17131 3 \frac{1}{3} 3 1
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