Linear Algebra Ch 1.6 — Applications of Linear Systems

Suppose an economy has only two sectors, Goods and Services. Each year, Goods sells 80% of its output to Services and keeps the rest, while Services sells 70% of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.

Find another set of equilibrium prices for the economy in Exercise 1. Suppose the same economy used Japanese yen instead of dollars to measure the value of the various sectors' outputs. Would this change the problem in any way? Discuss.

Consider an economy with three sectors, Chemicals & Metals, Fuels & Power, and Machinery. Chemicals sells 30% of its output to Fuels and 50% to Machinery and retains the rest. Fuels sells 80% of its output to Chemicals and 10% to Machinery and retains the rest. Machinery sells 40% to Chemicals and 40% to Fuels and retains the rest.

Suppose an economy has four sectors, Agriculture (A), Energy (E), Manufacturing (M), and Transportation (T). Sector A sells 10% of its output to E, 25% to M and retains the rest. Sector E sells 30% of its output to A, 35% to M, and 25% to T and retains the rest. Sector M sells 30% of its output to A, 15% to E, and 40% to T and retains the rest. Sector T sells 20% of its output to A, 10% to E, and 30% to M and retains the rest.

Balance the chemical equation using the vector equation approach discussed in this section. Boron sulfide reacts violently with water to form boric acid and hydrogen sulfide gas (the smell of rotten eggs). The unbalanced equation is: \(\text{B}_2 \text{S}_3 + \text{H}_2 \text{O} \rightarrow \text{H}_3 \text{BO}_3 + \text{H}_2 \text{S}\) [For each compound, construct a vector that lists the numbers of atoms of boron, sulfur, hydrogen, and oxygen.]

Balance the chemical equation using the vector equation approach. When solutions of sodium phosphate and barium nitrate are mixed, the result is barium phosphate (as a precipitate) and sodium nitrate. The unbalanced equation is: \(\text{Na}_3 \text{PO}_4 + \text{Ba}(\text{NO}_3)_2 \rightarrow \text{Ba}_3 (\text{PO}_4)_2 + \text{NaNO}_3\) [For each compound, construct a vector that lists the numbers of atoms of sodium (Na), phosphorus, oxygen, barium, and nitrogen. For instance, barium nitrate corresponds to (0, 0, 6, 1, 2).]

Balance the chemical equation using the vector equation approach. Alka-Seltzer contains sodium bicarbonate (\(\text{NaHCO}_3\)) and citric acid (\(\text{H}_3 \text{C}_6 \text{H}_5 \text{O}_7\)). When a tablet is dissolved in water, the following reaction produces sodium citrate, water, and carbon dioxide (gas): \(\text{NaHCO}_3 + \text{H}_3 \text{C}_6 \text{H}_5 \text{O}_7 \rightarrow \text{Na}_3 \text{C}_6 \text{H}_5 \text{O}_7 + \text{H}_2 \text{O} + \text{CO}_2\)

Balance the chemical equation using the vector equation approach. The following reaction between potassium permanganate (\(\text{KMnO}_4\)) and manganese sulfate in water produces manganese dioxide, potassium sulfate, and sulfuric acid: \(\text{KMnO}_4 + \text{MnSO}_4 + \text{H}_2 \text{O} \rightarrow \text{MnO}_2 + \text{K}_2 \text{SO}_4 + \text{H}_2 \text{SO}_4\) [For each compound, construct a vector that lists the numbers of atoms of potassium (K), manganese, oxygen, sulfur, and hydrogen.]

If possible, use exact arithmetic or rational format for calculations in balancing the following chemical reaction: \(\text{PbN}_6 + \text{CrMn}_2 \text{O}_8 \rightarrow \text{Pb}_3 \text{O}_4 + \text{Cr}_2 \text{O}_3 + \text{MnO}_2 + \text{NO}\)

The chemical reaction below can be used in some industrial processes, such as the production of arsene (\(\text{AsH}_3\)). Use exact arithmetic or rational format for calculations to balance this equation: \(\text{MnS} + \text{As}_2 \text{Cr}_{10} \text{O}_{35} + \text{H}_2 \text{SO}_4 \rightarrow \text{HMnO}_4 + \text{AsH}_3 + \text{CrS}_3 \text{O}_{12} + \text{H}_2 \text{O}\)