Consider the Solow growth model with government spending but without technological progress. Output \( Y \) is produced according to the production function \[ Y=F(K, L)=\sqrt[3]{K} \cdot \sqrt[3]{L^{2}} \] where \( K \) is the capital stock and \( L \) is the labour force. Labour force is equal to employment and total population and grows at the rate \( n \). Capital depreciates at the rate \( \delta \). The government spends ('consumes') resources without contributing to production or capital accumulation. Government purchases, \( G \), are financed with proportional income taxation. The government budget in per capita terms is balanced, \[ g=\tau y \] where \( y \equiv Y / L \) and \( g \equiv G / L \) are output and government purchases per capita, and \( \tau \in[0,1) \) denotes the proportional tax rate. Investment \( I \) and consumption \( C \) absorb, correspondingly, constant fractions \( s \) and \( (1-s) \) of disposable income. (a) [5 marks] Show that the function \( F(K, L) \) has constant returns to scale and diminishing marginal returns to capital.

IX. Please determine whether the set of vectors \( \left\{\begin{array}{lll}\alpha_{1} & \alpha_{2} & \alpha_{3}\end{array}\right\} \) is linearly dependent or not, where \( \alpha_{1}=\left(\begin{array}{lll}1 & 1 & 1\end{array}\right)^{T}, \alpha_{2}=\left(\begin{array}{llll}2 & 1 & 1\end{array}\right)^{T}, \alpha_{3}=\left(\begin{array}{lll}3 & 2 & 1\end{array}\right)^{T} \). (Write the answer for the questions on your answer sheet. 10 points)

Exorcise 3.11 . Let \( \mathrm{A}=\left(\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 3 & 6 & -5\end{array}\right) \). Find the matrix obtained from \( A \) by applying the following elementary row operation. \( \begin{array}{lll}\text { a. } R_{1} \leftrightarrow R_{3} & \text { b. } R_{2} \rightarrow R_{2}-4 R_{1} & \text { c. } R_{3} \rightarrow \frac{1}{3} R_{3}\end{array} \)

1-) \( \left(20\right. \) p) Let \( A=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 3 & 4 \\ 5 & 2 & 0\end{array}\right] \) be a matrix. Find \( \operatorname{det}(A) \), adj \( A \) and the inverse of the matrix if it exists.

4. If \( \quad\left[\begin{array}{cccccc}0 & 0 & -3 & -2 & 0 & 5 \\ -2 & 3 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 1 & -2 & 3 & 0 & 0 & 0 \\ -2 & 8 & 3 & 6 & 2 & 7 \\ 0 & 2 & 3 & 0 & 0 & 0\end{array}\right] \operatorname{compute} \operatorname{det}(A),(\operatorname{adj}(A))^{-1} \) and \( \operatorname{det}(\operatorname{adj}(A)) \)

Consider the Solow growth model with government spending but without technological progress. Output \( Y \) is produced according to the production function \[ Y=F(K, L)=\sqrt[3]{K} \cdot \sqrt[3]{L^{2}} \] where \( K \) is the capital stock and \( L \) is the labour force. Labour force is equal to employment and total population and grows at the rate \( n \). Capital depreciates at the rate \( \delta \). The government spends ('consumes') resources without contributing to production or capital accumulation. Government purchases, \( G \), are financed with proportional income taxation. The government budget in per capita terms is balanced, \[ g=\tau y \] where \( y \equiv Y / L \) and \( g \equiv G / L \) are output and government purchases per capita, and \( \tau \in[0,1) \) denotes the proportional tax rate. Investment \( I \) and consumption \( C \) absorb, correspondingly, constant fractions \( s \) and \( (1-s) \) of disposable income. (a) [5 marks] Show that the function \( F(K, L) \) has constant returns to scale and diminishing marginal returns to capital.