# An industrial system with three industries has the input-output matrix D and external demand matrix E below. Solve for the output matrix X in the equation X = DX + E. D=begin{bmatrix}0.1 & 0.3&0.2 0.0 & 0.2&0.30.4&0.1&0.1 end{bmatrix} text{ and } E=begin{bmatrix}3000 35008500 end{bmatrix}

An industrial system with three industries has the input-output matrix D and external demand matrix E below. Solve for the output matrix X in the equation
X = DX + E.
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2abehn
Step 1
Given:
The equation X=DX+E
and the matrices:

Step 2
To solve the equation
X=DX+E
Subtract DX from both sides
X-DX=E
$⇒\left(I-D\right)X=E$
Pre-multiply both sides by $\left(I-D{\right)}^{-1}$
$⇒X=\left(I-D{\right)}^{-1}E\dots \left(1\right)$
Step 3
To simplify further, compute the matrix (I-D), and then its
Jeffrey Jordon