Linear Algebra & Applications

2201NSC

Week 1 - Intro & Simultaneous Equations

Problem 1

Write augmented matrices for the following sets of equations, and solve

(a) $\begin{aligned} x_1 - 2x_2 &= 5 \\ 3x_1 + x_2 &= 1 \\ & \end{aligned}$ (c) $\begin{aligned} x_1 + x_2 &= 1 \\ x_1 - x_2 &= 1 \\ -x_1 + 3x_2 &= 3 \\ & \end{aligned}$
(b) $\begin{aligned} 3x_1 + 2x_2 + x_3 &= 0\\ -2x_1 + x_2 - x_3 &= 2 \\ 2x_1 - x_2 + 2x_3 &= -1 \end{aligned}$ (d) $\begin{aligned} x_1 + x_2 + x_3 &= 1 \\ x_1 - x_2 - x_3 &= 1 \end{aligned}$



Problem 1 (cont.)

(a) $\begin{aligned} x_1 - 2x_2 &= 5 \\ 3x_1 + x_2 &= 1 \\ & \end{aligned}$









Problem 2

Write sets of equations for the following augmented matrices, and solve

(a) $\, \begin{pmatrix} 5 & -2 & 1 & 3 \\ 2 & 3 & -4 & 0 \end{pmatrix} $ (c) $ \,\begin{pmatrix} 2 & 1 & 4 & 0 \\ 4 & 2 & 3 & 5 \\ 5 & 2 & 6 & 1 \end{pmatrix} $
(b) $\,\begin{pmatrix} 2 & 1 & 3 \\ 1 & -1 & 0 \\ 4& -1 & 3 \end{pmatrix} $ (d) $ \,\begin{pmatrix} 3 & 2 & 8 \\ 1 & 5 & 7 \end{pmatrix} $



Problem 2 (cont.)

(a) $\, \begin{pmatrix} 5 & -2 & 1 & 3 \\ 2 & 3 & -4 & 0 \end{pmatrix} $









Problem 3

Determine whether the following echelon forms correspond to (i) inconsistent systems, (ii) systems with a unique solution, or (iii) system with a family of solutions

(a) $\, \left(\begin{array}{cc|c} 1 & 2 & 4 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right) $ (c) $ \,\left(\begin{array}{cc|c} 1 & 3 & 1\\ 0 & 1 & -1 \\ 0 & 0 & 0\end{array}\right) $
(b) $\, \left(\begin{array}{ccc|c} 1 & -2 & 2 & -2 \\ 0 & 1 & -1 & 3 \\ 0 & 0 & 1 & 2 \end{array}\right) $ (d) $\, \left(\begin{array}{ccc|c} 1 & -2 & 4 & 1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right) $



Problem 3 (cont.)

(a) $\, \left(\begin{array}{cc|c} 1 & 2 & 4 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{array}\right) $









Problem 4

Balance the chemical equations

  1. $ x_1\text{C}_4\text{H}_{10} + x_2\text{O}_2\ \rightarrow\ x_3\text{CO}_2 + x_4\text{H}_2\text{O}$
  2. $ x_1\text{Cu} + x_2\text{HNO}_3\ \rightarrow\ x_3\text{Cu(NO}_3\text{)}_2+ x_4\text{NO} + x_5\text{H}_2\text{O}$
  3. $ x_1\text{P}_4\text{O}_{10} + x_2\text{H}_2\text{O}\ \rightarrow\ x_3\text{H}_3\text{PO}_4$







Problem 5

Port Ashfield has a local railroad supporting its coal mine and electricity power plant. Mining $\$100$ of coal costs $\$25$ in electricity and $\$25$ in rail access. Generating $\$100$ of electricity costs $\$65$ in coal, $\$5$ in electricity and $\$5$ in rail access. Finally, in charging $\$100$ for rail access, the railroad company spends $\$55$ on coal and $\$10$ in electricity.

Over a given period, the Port Ashfield receives purchase orders for $\$2M$ of coal and $\$1M$ of electricity. How much must each industry produce (measured in $\$$) so that it can meet all demands.



Problem 5 (cont.)












Problem 6

A medieval village has a dairy farmer whose farm produces 500 litres of milk each week, a corn farmer who produces 100 kg each week on average, and a cheese maker who produces 10 kg of cheese each week. They exchange goods with each other through the village market. Each week, the diary farmer trades 150 litres for 20kg of corn and 4 kg of cheese; the corn farmer trades 30 kg of corn for 3 kg of cheese and 100 litres of milk; and the cheese maker takes the remaining 50 litres of milk and 10 kg of corn in exchange for the 7kg of cheese provided.

One day, a foreigner arrives at the village market, paying 100 coins for 1kg of cheese, 2 kg of corn, and 3 litres of milk. How should these coins be distributed among the diary farmer, corn farmer and cheese maker, so that each is happy with the amount they receive?



Problem 6 (cont.)












Problem 7

Here we consider a simplified version of Uber's system for calculating rides charges. We will consider only flagfall per trip, rate per minute, and rate per kilometre. The following is a table of Uber costs to a small business over one month.

Who # trips Total time Total distance Total bill
Michelle 6 3 hr 60 km $161.70
Zach 3 2 hr 60 km $129.75
Anna 10 5 hr 200 km $394.50

How much should the company allow in next month's budget, if it expects 20 trips, lasting 8 hours, covering total distance of 300km?


Problem 7 (cont.)