91Ó°ÊÓ

Consider an open economy with consumption matrix $$ C=\left[\begin{array}{lll} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{8} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{8} \end{array}\right] $$ If the open sector demands the same dollar value from each product-producing sector, which such sector must produce the greatest dollar value to meet the demand?

Determine the product by inspection. $$\left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 4 \end{array}\right]\left[\begin{array}{rrr} 4 & -1 & 3 \\ 1 & 2 & 0 \\ -5 & 1 & -2 \end{array}\right]\left[\begin{array}{rrr} -3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{array}\right]$$

Find the inverse of $$ \left[\begin{array}{rr} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] $$

Consider an open economy with consumption matrix $$ C=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & 0 \end{array}\right] $$ Show that the Leontief equation \(\mathbf{x}-C \mathbf{x}=\mathbf{d}\) has a unique solution for every demand vector \(\mathbf{d}\) if \(c_{21} c_{12}<1-c_{11}\).

Find the inverse of $$ \left[\begin{array}{ll} \frac{1}{2}\left(e^{x}+e^{-x}\right) & \frac{1}{2}\left(e^{x}-e^{-x}\right) \\\ \frac{1}{2}\left(e^{x}-e^{-x}\right) & \frac{1}{2}\left(e^{x}+e^{-x}\right) \end{array}\right] $$

Solve the linear systems together by reducing the appropriate augmented matrix. \(x_{1}-5 x_{2}=b_{1}\) \(3 x_{1}+2 x_{2}=b_{2}\) (i) \(b_{1}=1, \quad b_{2}=4\) (ii) \(b_{1}=-2, \quad b_{2}=5\)

Use the inversion algorithm to find the inverse of the given matrix, if the inverse exists. $$\left[\begin{array}{ll} 1 & 4 \\ 2 & 7 \end{array}\right]$$

In each part, find the solution set of the linear equation by using parameters as necessary. (a) \(7 x-5 y=3\) (b) \(-8 x_{1}+2 x_{2}-5 x_{3}+6 x_{4}=1\)

Find \(A^{2}, A^{-2},\) and \(A^{-k}\) (where \(k\) is any integer) by inspection. $$A=\left[\begin{array}{cc} 1 & 0 \\ 0 & -2 \end{array}\right]$$

Write a balanced equation for the given chemical reaction. $$\mathrm{C}_{3} \mathrm{H}_{8}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} \quad \text { (propane combustion) }$$