Answer:
Part a: In order to meet the external demand of 100 units of Product 1, 120 units of Product 2 and 150 units of Product 3, the total production of 147.06 units of Product 1, 171.27 units of Product 2 and 184.29 units of Product 3 are to be produced.
Part b: The individual entry of 3rd column (I-M)^-1 signifies the role of the demand of 3rd product for total estimation of product 1, product 2 and product 3 respectively.
Explanation:
The matrix form of the equation is given as
[tex]\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] =\left[\begin{array}{ccc}0.02&.0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] +\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] _{external}[/tex]
where
[tex]\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] _{external}=\left[\begin{array}{c}100\\120\\150\end{array}\right][/tex]
so the equation now becomes
[tex]\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] =\left[\begin{array}{ccc}0.02&.0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] +\left[\begin{array}{c}100\\120\\150\end{array}\right][/tex]
From here it is given that
[tex]P=MP+\Delta[/tex]
Or
[tex]P-MP=\Delta\\P(I-M)=\Delta\\P=(I-M)^{-1}\Delta[/tex]
Here
[tex]P=\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] \\M=\left[\begin{array}{ccc}0.02&.0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \\\Delta=\left[\begin{array}{c}100\\120\\150\end{array}\right][/tex]
So now I-M is given as
[tex]\\I-M=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] -\left[\begin{array}{ccc}0.02&0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \\ I-M=\left[\begin{array}{ccc}1-0.02&0-0.15&0-0.10\\0-0.14&1-0.05&0-0.12\\0-0.14&0-0.08&1-0\end{array}\right] \\I-M=\left[\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right][/tex]
Now the inverse is calculated as
[tex](I-M)^{-1}=\frac{1}{det(I-M)}Adj(I-M)[/tex]
So the adjoint of (I-M) is calculated as
[tex]adj(I-M)=adj(\left[\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right])\\adj(I-M)=\left[\begin{array}{ccc}0.9404&0.1580 & 0.1130\\0.1568&0.9660& 0.1316\\0.1442 &0.0994 & 0.9100\end{array}\right]\\[/tex]
Also the determinant is given as
[tex]|I-M|=\left|\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right|\\|I-M|=0.8837[/tex]
So the inverse is given as
[tex](I-M)^{-1}=\frac{1}{det(I-M)}Adj(I-M)[/tex]
[tex](I-M)^{-1}=\frac{1}{0.8837}\left[\begin{array}{ccc}0.9404&0.1580 & 0.1130\\0.1568&0.9660& 0.1316\\0.1442 &0.0994 & 0.9100\end{array}\right]\\\\(I-M)^{-1}=\left[\begin{array}{ccc}1.0642 & 0.1788 & 0.1279\\ 0.1774 &1.0932 & 0.1489\\ 0.1632 & 0.1125 & 1.0298\end{array}\right]\\[/tex]
So the total demand of each product to meet the external demand is given as
[tex]P=(I-M)^{-1}\Delta[/tex]
[tex]\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right]=\left[\begin{array}{ccc}1.0642 & 0.1788 & 0.1279\\ 0.1774 &1.0932 & 0.1489\\ 0.1632 & 0.1125 & 1.0298\end{array}\right]\left[\begin{array}{c}100\\120\\150\end{array}\right]\\\\\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right]=\left[\begin{array}{c}147.06\\ 171.27 \\184.29\end{array}\right]\\[/tex]
So in order to meet the external demand of 100 units of Product 1, 120 units of Product 2 and 150 units of Product 3, the total production of 147.06 units of Product 1, 171.27 units of Product 2 and 184.29 units of Product 3 are to be produced.
Part b
The individual entry of 3rd column (I-M)^-1 signifies the role of the demand of 3rd product for total estimation of product 1, product 2 and product 3 respectively.
