Math  /  Algebra

QuestionFind the production schedule for the technology matrix and demand vector A=[0.30.70.20.10.20.10.81.20.4],D=[754]A=\left[\begin{array}{lll} 0.3 & 0.7 & 0.2 \\ 0.1 & 0.2 & 0.1 \\ 0.8 & 1.2 & 0.4 \end{array}\right] \quad, \quad D=\left[\begin{array}{l} 7 \\ 5 \\ 4 \end{array}\right]

Studdy Solution
Calculate the inverse of the matrix (IA) (I - A) and then multiply it by D D to find X X :
First, compute IA I - A :
I=[100010001] I = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]
IA=[100010001][0.30.70.20.10.20.10.81.20.4]=[0.70.70.20.10.80.10.81.20.6] I - A = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] - \left[\begin{array}{ccc} 0.3 & 0.7 & 0.2 \\ 0.1 & 0.2 & 0.1 \\ 0.8 & 1.2 & 0.4 \end{array}\right] = \left[\begin{array}{ccc} 0.7 & -0.7 & -0.2 \\ -0.1 & 0.8 & -0.1 \\ -0.8 & -1.2 & 0.6 \end{array}\right]
Next, calculate the inverse of (IA) (I - A) . For simplicity, let's denote B=(IA) B = (I - A) and find B1 B^{-1} .
Finally, multiply B1 B^{-1} by D D to find X X :
X=B1D X = B^{-1}D
Perform the matrix multiplication to find the production vector X X .

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