Math Challenges

Submissions for Problem #47

Problem #47

Solve for production vector x given consumption matrix C and demand d.

\[ \displaylines{\mathbf{x}=C\mathbf{x}+\mathbf{d}\\ C=\begin{pmatrix}.0 & .5\\ .6 & .2\end{pmatrix},\text{ }\mathbf{d}=\begin{pmatrix}50\\ 30\end{pmatrix}} \]

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zapwai

Solution: \( \displaylines{x-Cx=\left(I-C\right)x=d\implies x=\left(I-C\right)^{-1}d\\ I-C=\begin{pmatrix}1 & -0.5\\ -0.6 & 0.8\end{pmatrix}\implies\left(I-C\right)^{-1}=\begin{pmatrix}1.6 & 1\\ 1.2 & 2\end{pmatrix}\\ x=\begin{pmatrix}1.6 & 1\\ 1.2 & 2\end{pmatrix}\begin{pmatrix}50\\ 30\end{pmatrix}=\begin{pmatrix}110\\ 120\end{pmatrix}} \)