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Home Question Bank Other Archive 2024 December 24

Other Questions from Dec 24,2024

Browse the Other Q&A Archive for Dec 24,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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1. Representar en un diagrama de Venn el conjunto \( [A-(A \cap B)]^{C} \). Considere a) \( A \) subconjunto de \( B \) b) \( B \) subconjunto de \( A \) 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 \( \operatorname{det}(\operatorname{adj}(A)) \) 15. (a) (i) Let \( \quad(L, \leq) \quad \) be a lattice. For any \( a, b \in L \), \( a \leq b \Leftrightarrow a \wedge b=a \Leftrightarrow a \vee b=b \). Calcular la proyección del vector \( \vec{u}=(-3,2,1) \) sobre el vector \( \vec{v}=(-1,2,4) \) a. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) b. \( \vec{w}=\left(-\frac{11}{21}, \frac{22}{21}, \frac{44}{21}\right) \) c. \( \vec{w}=\left(-\frac{11}{21},-\frac{22}{21}, \frac{44}{21}\right) \) d. \( \vec{w}=\left(\frac{11}{21}, \frac{22}{21},-\frac{44}{21}\right) \) 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. Q4: Given the matrix \( A=\left(\begin{array}{ccc}7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 3\end{array}\right) \) Where \( X 1=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right) \quad, X 2=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right), X 3=\left(\begin{array}{c}1 \\ -2 \\ 1\end{array}\right) \) Find: 1) the rank of \( A \) ? 2) the eigen values and eigen vectors? 3) satisfy \( P^{\prime} / A P=\operatorname{Diag}\left(\begin{array}{lll}\lambda_{1} & \lambda_{2} & \lambda_{3}\end{array}\right) \) or \( N o \) ?
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