The given economy can be defined by the following system of equations with the final demand matrix (\mathbf{d}):

[ \begin{array}{ c|c c c |c } & \text{Agriculture} & \text{Manufactures} & \text{Services} & \text{Total} \ \hline \text{Agriculture} & 60 & 180 & 0 & 120 \ \text{Manufactures} & 30 & 60 & 30 & 180 \ \text{Services} & 0 & 40 & 10 & 100 \ \text{Final Demand} & 150 & 160 & 180 & \end{array} ]

First, let's form the input-output matrix (\mathbf{C}):

[ C = \begin{pmatrix} 0.30 & 0.60 & 0.00 \ 0.15 & 0.20 & 0.05 \ 0.00 & 0.27 & 0.05 \end{pmatrix} ]

Now define the vector of total outputs (\mathbf{x}) and the vector of final demands (\mathbf{d}):

[ \mathbf{x} = \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 150 \ 160 \ 180 \end{pmatrix} ]

The system of equations to solve is:

[ \mathbf{x} = \mathbf{C}\mathbf{x} + \mathbf{d} ]

This can be rewritten as:

[ (\mathbf{I} - \mathbf{C})\mathbf{x} = \mathbf{d} ]

[ \mathbf{I} - \mathbf{C} = \begin{pmatrix} 1 - 0.30 & -0.60 & 0.00 \ -0.15 & 1 - 0.20 & -0.05 \ 0.00 & -0.27 & 1 - 0.05 \end{pmatrix}

\begin{pmatrix} 0.70 & -0.60 & 0.00 \ -0.15 & 0.80 & -0.05 \ 0.00 & -0.27 & 0.95 \end{pmatrix} ]

Now solve for (\mathbf{x}):

[ \begin{pmatrix} 0.70 & -0.60 & 0.00 \ -0.15 & 0.80 & -0.05 \ 0.00 & -0.27 & 0.95 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix}

\begin{pmatrix} 150 \ 160 \ 180 \end{pmatrix} ]

By performing Gaussian elimination or using matrix inversion for ((\mathbf{I} - \mathbf{C})^{-1}) to solve for (\mathbf{x}):

[ (\mathbf{I} - \mathbf{C})^{-1} \approx \begin{pmatrix} 1.66071 & 1.25607 & 0.06690 \ 0.16677 & 1.33915 & 0.07057 \ 0.07986 & 0.35041 & 1.03829 \end{pmatrix} ]

Now multiplying the inversed matrix with the final demand vector (\mathbf{d}):

[ \mathbf{x} = \begin{pmatrix} 1.66071 & 1.25607 & 0.06690 \ 0.16677 & 1.33915 & 0.07057 \ 0.07986 & 0.35041 & 1.03829 \end{pmatrix} \begin{pmatrix} 150 \ 160 \ 180 \end{pmatrix} ]

[ = \begin{pmatrix} 1.66071 \cdot 150 + 1.25607 \cdot 160 + 0.06690 \cdot 180 \ 0.16677 \cdot 150 + 1.33915 \cdot 160 + 0.07057 \cdot 180 \ 0.07986 \cdot 150 + 0.35041 \cdot 160 + 1.03829 \cdot 180 \end{pmatrix} ]

[ = \begin{pmatrix} 249.1075 + 200.9712 + 12.0420 \ 25.0155 + 214.2639 + 12.7038 \ 11.9790 + 56.0656 + 186.8882 \end{pmatrix} ]

[ \approx \begin{pmatrix} 396.14 \ 300.13 \ 235.75 \end{pmatrix} ]

Thus, the new output levels are approximately:

• Agriculture: (396.14)
• Manufactures: (300.13)
• Services: (235.75)