Study sheet: Linear Equations and Matrix Algebra

📋 Course Outline

1. Linear Equations and Slopes
2. Developing Line Equations
3. Business Cost and Break-Even Models
4. Matrix Basics and Operations
5. Scalar and Matrix Multiplication
6. Matrix Inverses and Row Operations
7. Linear Systems and Gaussian Elimination
8. Word Problems and Markov Chains
9. Markov Chain Principles
10. Linear Programming Model Structure
11. Linear Programming Formulation
12. Graphical Solution Method
13. Slack and Surplus in Linear Programs
14. The Simplex Algorithm
15. Linear Programming Applications
16. Interest and Time Value of Money
17. Ordinary Annuities and Future Value
18. Sinking Funds and Loan Payments

📖 1. Linear Equations and Slopes

🔑 Key Concepts & Definitions

• Algebraic expression : a mathematical statement indicating that numerical quantities are linked by mathematical operations.
• Equation : a mathematical statement indicating that two algebraic expressions are equal.
• Linear equation : has variables of degree one and contains only constants or constants multiplied by one variable to the first power.
• Slope : The slope of a line measures the change in the value of $Y$ relative to a one-unit change in the value of $X$ and measures the line's steepness.

📝 Essential Points

📐 Formula — The general form of a linear equation is $Y=mx+b$, where $m$ is the slope, $b$ is the Y-intercept, $Y$ is the dependent variable, and $X$ is the independent variable.

💡 Memory Hook

Linear terms keep a constant slope; products, powers, and quotients of variables do not.

📖 2. Developing Line Equations

★ Must-know

• A line can be developed using the slope-intercept form, the slope-point form, or the two-point form.

📐 Formula — The slope-intercept equation of a line with slope $m$ and Y-intercept $b$ is $Y=mX+b$.

📐 Formula — The slope-point equation of a non-vertical line with slope $m$ passing through $(X_1,Y_1)$ is $Y-Y_1=m(X-X_1)$.

📌 A line is horizontal when its two points have the same Y-coordinate and has equation $Y=\text{constant}$, whereas it is vertical when its points have the same X-coordinate and has equation $X=\text{constant}$.

📌 Two non-vertical lines are parallel when they have the same slope and perpendicular when the product of their slopes is $-1$.

Further detail

📐 Formula — For two points $(X_1,Y_1)$ and $(X_2,Y_2)$, the slope is $m=\frac{Y_2-Y_1}{X_2-X_1}$ when $X_1\ne X_2$, after which the point-slope form gives the line equation.

💡 Memory Hook

Slope-intercept → point-slope → two-point form

📖 3. Business Cost and Break-Even Models

🔑 Key Concepts & Definitions

• Total cost : Total cost is the sum of total variable cost and total fixed cost, expressed as $TC=VC+FC$.
• Break-even point : the production quantity or revenue level at which total revenue equals total cost and the company makes neither profit nor loss.

★ Must-know

📐 Formula — For unit selling price $P$ and quantity $Q$, total revenue is $TR=PQ$ and profit is $\Pi=TR-TC$.

📌 In a linear cost model, marginal cost equals unit variable cost, average variable cost remains constant, and average fixed cost decreases as quantity increases.

📐 Formula — For a manufacturing company, break-even quantity is $Q_e=\frac{FC}{P-VC}$, where $FC$ is fixed cost, $P$ is unit selling price, and $VC$ is unit variable cost.

Further detail

📌 Increasing fixed cost or unit variable cost increases the break-even quantity, whereas increasing selling price decreases the break-even quantity.

📐 Formula — For a retail business with variable-cost proportion $m$ of sales revenue, break-even revenue is $BER=\frac{FC}{1-m}$.

💡 Memory Hook

Fixed cost, variable cost, and price determine the break-even quantity or revenue.

📖 4. Matrix Basics and Operations

🔑 Key Concepts & Definitions

• Matrix : a rectangular array of real numbers arranged in $m$ rows and $n$ columns.
• Matrix types : The main matrix types presented are vector matrices, square matrices, zero matrices, identity matrices, and scalar matrices.
• Identity matrix : An identity matrix is a square matrix with ones on the primary diagonal and zeros elsewhere, and multiplying a conformable matrix by it leaves that matrix unchanged.

★ Must-know

• The element $a_{ij}$ of a matrix is located in row $i$ and column $j$, and the matrix has dimension $m\times n$ when it contains $m$ rows and $n$ columns.

📌 Two matrices are conformable for addition or subtraction only when they have the same dimensions, and corresponding elements are then added or subtracted.

Further detail

📌 Multiplying a matrix by a scalar multiplies every element by that scalar and produces a matrix with the same dimension.

📌 For conformable matrices and real scalars, matrix addition is commutative and associative, and scalar multiplication distributes over matrix addition.

💡 Memory Hook

Define the array → classify its type → add or multiply conformable matrices

📖 5. Scalar and Matrix Multiplication

🔑 Key Concepts & Definitions

• Conformable matrices : Two matrices are conformable for multiplication when the number of columns in the first matrix equals the number of rows in the second matrix.

★ Must-know

📌 Scalar multiplication of matrices satisfies X A = A X for a real scalar X.

📌 Scalar multiplication is distributive over scalar addition and matrix addition: (X + Y)A = XA + YA and X(A + B) = XA + XB.

• If A has dimension n × m and B has dimension m × p, then AB has dimension n × p.

📌 Matrix multiplication is associative and distributive, but it is not generally commutative: AB does not generally equal BA.

Further detail

📌 Scalar multiplication is associative: X(YA) = XY(A).

📌 For matrices A and B, AB may be zero even when neither A nor B is a zero matrix, and AB = AC does not generally imply B = C.

💡 Memory Hook

SAD: scalar laws, admissibility, distributivity

📖 6. Matrix Inverses and Row Operations

🔑 Key Concepts & Definitions

• Matrix inverse : For a square matrix A, A⁻¹ is its inverse if and only if AA⁻¹ = A⁻¹A = I.

📝 Essential Points

📌 An invertible, or nonsingular, matrix has an inverse, whereas a singular matrix has no inverse.

• Only square matrices can have inverses, and whenever an inverse exists it is unique.

• 🔄 The inverse-finding process is:

  1. Augment A with I
  2. Apply elementary row operations
  3. Transform A into I
  4. Read A⁻¹ on the right

• The elementary row operations are:

  • Interchanging two rows
  • Multiplying a row by a nonzero real number
  • Adding a multiple of one row to another row

💡 Memory Hook

Augment → reduce → read the inverse

📖 7. Linear Systems and Gaussian Elimination

★ Must-know

• The inverse method for AX = B changes the system into matrix form, finds the inverse of the coefficient matrix, and multiplies that inverse by the constant vector to obtain X = A⁻¹B.

📌 The inverse method applies only when the coefficient matrix is square and invertible, and it does not distinguish between no-solution and infinitely-many-solution cases.

• Gaussian elimination augments the coefficient matrix with the constant vector and applies elementary row operations until the coefficient matrix is in identity form or reveals the solution type.

📌 A row of zeros with a nonzero constant indicates no solution, an identity-form coefficient matrix gives a unique solution, and a consistent reduced form with free variables indicates infinitely many solutions.

Further detail

• Every system of linear equations has either no solution, exactly one solution, or infinitely many solutions.

💡 Memory Hook

Unique solution versus no solution versus infinitely many solutions

📖 8. Word Problems and Markov Chains

🔑 Key Concepts & Definitions

• Markov chain : Andrew Markov, around 1907 — A probabilistic model for studying how a system evolves over repeated time periods using transition probabilities between states.

★ Must-know

• To solve a word problem, represent an unknown with a letter, express other unknowns in terms of it, translate the information into algebraic equations, solve the equations, and check the result against the statement.

Further detail

• In the carpeting example, using 1,500 square yards of wool and 1,800 square yards of nylon requires 20X + 30Y = 1,500 and 40X + 30Y = 1,800, giving X = 15 superior-grade rolls and Y = 40 quality-grade rolls.

• Markov chains can be used to predict the probability that a system will be in a particular state at a given time and to study its long-run equilibrium or steady state.

💡 Memory Hook

Represent → translate → solve → check

📖 9. Markov Chain Principles

🔑 Key Concepts & Definitions

• Markov chain : Andrew Markov, around 1907 — a stochastic forecasting model that describes the evolution of a system across successive time periods using transition probabilities
• Transition probability : the likelihood that a system currently in state i will move to state j during the next period

★ Must-know

📌 The state of a system in a given period depends on its state in the preceding period and on the transition probabilities.

📐 Formula — Future state vectors are calculated by multiplying the previous state vector by the transition matrix: $V(n)=V(n-1)P$.

• For the two-store example, the long-run distribution is two-thirds of customers at store 1 and one-third at store 2.

Further detail

• The assumptions of a Markov chain are:

  • constant transition probabilities over time
  • at most one system change per period
  • regular transition periods

• For the two-store example, the initial distribution is (0.8, 0.2), the transition matrix is $P=\begin{pmatrix}0.9&0.1\\0.2&0.8\end{pmatrix}$, and the first two predicted distributions are (0.76, 0.24) and (0.732, 0.268).

💡 Memory Hook

Current state + transition probabilities → future and steady-state predictions

📖 10. Linear Programming Model Structure

🔑 Key Concepts & Definitions

• Linear programming : an optimization method for allocating scarce resources subject to limiting conditions expressed as inequalities or equations
• Decision variables : the unknown controllable quantities that the decision maker must determine
• Constraints : restrictions that limit the feasibility or attainability of a proposed course of action, such as scarce resources, legal requirements, contractual requirements, forecasts, orders, or company policies
• Parameters : fixed numerical values specifying the effect of each decision variable on the objective and on the constraints

📝 Essential Points

• A linear programming objective may involve:

  • maximization of profit
  • maximization of revenue
  • maximization of sales
  • maximization of market share
  • minimization of cost
  • minimization of time
  • minimization of distance

• The main assumptions are:

  • linearity
  • divisibility
  • certainty
  • non-negativity

💡 Memory Hook

ODCP: objective, decision variables, constraints, parameters

📖 11. Linear Programming Formulation

★ Must-know

• 🔄 Model formulation involves:
  1. defining the problem
  2. identifying the decision variables
  3. specifying the objective function
  4. identifying the constraints

📐 Formula — The microcomputer model maximizes $Z=60X_1+50X_2$ subject to $4X_1+10X_2\le100$, $2X_1+X_2\le22$, $3X_1+3X_2\le39$, and $X_1,X_2\ge0$.

📌 Non-negativity constraints require every decision variable to be zero or positive because negative quantities are unrealistic.

Further detail

• For the microcomputer problem, the objective is to maximize weekly profit by choosing the quantities of two computer types to produce.

📐 Formula — The switching-device model maximizes $Z=8.25X_1+7.50X_2+7.80X_3$ subject to two assembly-time constraints of 450 minutes each, minimum production of 20 units for each model, and non-negativity.

💡 Memory Hook

Define the problem → variables → objective → constraints

📖 12. Graphical Solution Method

🔑 Key Concepts & Definitions

• Slack : the amount of a scarce resource that remains unused under a given solution and equals the available amount minus the amount used

★ Must-know

📌 The graphical solution method applies only to linear programming problems with two decision variables, whereas the algebraic simplex method can handle problems with more than two variables.

• The graphical method solves a problem by plotting each constraint, identifying the common feasible region, and selecting the point that gives the maximum benefit or minimum cost.

📌 For a linear programming problem with an optimum, at least one optimal solution occurs at an extreme or corner point of the feasible region.

• To identify the graphical optimum, determine the coordinates of each corner point, substitute them into the objective function, and select the point with the highest or lowest value as required.

Further detail

• For the microcomputer model, the optimal solution is to produce 9 units of type 1 and 4 units of type 2, giving a maximum profit of Br. 740.

• In the microcomputer solution, assembly time has 24 hours of slack, while inspection time and storage space are binding constraints with zero slack.

💡 Memory Hook

Plot constraints → find feasible region → test corner points → choose the best value

📖 13. Slack and Surplus in Linear Programs

🔑 Key Concepts & Definitions

• Binding constraint : a constraint with zero slack that limits or binds the solution
• Surplus : the amount by which an optimal solution exceeds the required minimum in a greater-than-or-equal-to constraint

📝 Essential Points

• In the microcomputer example, producing 9 units of type 1 and 4 units of type 2 gives a profit of Br. 740 and leaves 24 assembly hours unused, while inspection time and storage space are binding constraints.

💡 Memory Hook

Slack belongs to ≤ constraints, whereas surplus belongs to ≥ constraints.

📖 14. The Simplex Algorithm

🔑 Key Concepts & Definitions

• Simplex method : an iterative algebraic procedure that starts with a feasible but nonoptimal solution and improves it until no further improvement is possible

★ Must-know

📌 A basic solution is a solution obtained from the constraint equations, whereas a basic feasible solution also satisfies all constraints, including non-negativity constraints.

• For a maximization problem with ≤ constraints, the simplex procedure converts the model to standard form, develops the initial tableau, selects entering and leaving variables, performs row operations around the pivot, recomputes Cj − Zj, and repeats until no positive Cj − Zj remains.

📌 A maximization simplex solution is optimal when the Cj − Zj row contains only zeros and negative numbers.

Further detail

• For a model with n variables and m constraints, the number of possible basic solutions is represented as $\binom{n+m}{m}$.

💡 Memory Hook

Standard form → tableau → pivot → optimality test.

📖 15. Linear Programming Applications

📝 Essential Points

• For the lawn-mower problem, the model is $Z_{\max}=45X_1+70X_2$ subject to assembly, packing, and engine constraints, and the optimal plan is 24 push-type mowers and 42 self-propelled mowers with profit Br. 4020 and 9 unused engines.

• For the bentwood furniture problem, the optimal plan is 13 rocking chairs and 4 coffee tables, producing a maximum profit of Br. 648 and leaving 8 unused woodworking hours.

• For the tire-machine problem, the minimum operating cost of Br. 3600 is obtained by producing 40 tires on machine I and 20 tires on machine II.

💡 Memory Hook

Resource constraints → feasible region → optimal production plan → profit or cost.

📖 16. Interest and Time Value of Money

🔑 Key Concepts & Definitions

• Interest : the price paid for using a sum of money over a period of time and is usually computed as a percentage of the principal
• Present value : the amount that must be invested now to accumulate a specified future amount after a given number of compound periods
• Annuity : a sequence of equal periodic payments made weekly, monthly, quarterly, annually, semiannually, or at another fixed interval

★ Must-know

📌 Simple interest is calculated only on the initial principal, whereas compound interest is calculated on the principal plus previously accrued interest.

📐 Formula — Simple interest is calculated by $I=Prt$, and the maturity amount is $A=P(1+rt)$, where P is principal, r is the annual rate, and t is time in years.

📐 Formula — For compound interest, the amount after n conversion periods is $A=P(1+i)^n$, where $i=\frac{r}{m}$ and $n=mt$.

📐 Formula — The present value of a compound amount is calculated by $P=A(1+i)^{-n}$.

📐 Formula — The effective annual rate for a nominal rate r compounded m times per year is $r_e=\left(1+\frac{r}{m}\right)^m-1$.

📌 An ordinary annuity makes payments at the end of each payment period, whereas an annuity due makes payments at the beginning of each payment period.

Further detail

📐 Formula — Under continuous compounding, the amount is calculated by $A=Pe^{rt}$, where P is principal, r is the annual rate, and t is time in years.

💡 Memory Hook

Simple interest earns on principal only, whereas compound interest earns on principal plus accumulated interest.

📖 17. Ordinary Annuities and Future Value

🔑 Key Concepts & Definitions

• Ordinary annuity : a series of equal periodic payments in which each payment is made at the end of the period

★ Must-know

📌 An ordinary annuity consists of equal periodic payments made at the end of each payment period, whereas an annuity due has payments at the beginning of each period.

📐 Formula — The future value of an ordinary annuity is $A=R\frac{(1+i)^n-1}{i}$, where R is the periodic payment, i is the interest rate per payment period, and n is the number of payment periods.

Further detail

• Depositing Br. 100 at the end of each quarter for one year at 4% compounded quarterly produces a future value of Br. 406.04.

💡 Memory Hook

Payment → accumulation → future value

📖 18. Sinking Funds and Loan Payments

🔑 Key Concepts & Definitions

• Sinking fund : a fund receiving equal periodic payments in order to accumulate a definite amount of money on a specified date
• Present value : the sum of the present values of all payments discounted to the beginning of the annuity term
• Amortization : retiring a debt over a specified time through equal periodic payments that include compound interest
• Derivative : the general expression for the slope or instantaneous rate of change of a function

📝 Essential Points

📐 Formula — The periodic sinking-fund deposit is $R=A\frac{i}{(1+i)^n-1}$, where A is the target amount, i is the rate per payment period, and n is the number of deposits.

📐 Formula — The present value of an ordinary annuity is $P=R\frac{1-(1+i)^{-n}}{i}$.

📐 Formula — The amortization payment is $R=P\frac{i}{1-(1+i)^{-n}}$, where P is the present value of the loan, i is the rate per period, and n is the number of payments.

📐 Formula — The derivative of a constant function is $\frac{d}{dx}(K)=0$.

📐 Formula — For a power function, the power rule gives $\frac{d}{dx}(x^n)=nx^{n-1}$.

📌 If marginal revenue exceeds marginal cost, producing the next unit increases total profit; if marginal revenue is less than marginal cost, the next unit should not be produced.

📌 Profit is maximized at a critical production level where marginal revenue equals marginal cost, equivalently where $P'(X)=R'(X)-C'(X)=0$.

• The second-derivative test classifies a critical point by setting the first derivative to zero and evaluating the second derivative: a negative second derivative indicates a local maximum, a positive second derivative indicates a local minimum, and a zero second derivative makes the test inconclusive.

📌 A critical point where the first derivative is zero must be tested with the second derivative: a positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum.

📐 Formula — For weekly demand price $P(X)=10-0.001X$, revenue is $R(X)=XP(X)=10X-0.001X^2$ and profit is $P_f(X)=R(X)-C(X)=8X-0.001X^2-5000$ when cost is $C(X)=5000+2X$.

• The revenue function reaches its maximum at a production level of 5,000 transistor radios per week, producing a maximum revenue of 25,000 Br.

• The profit function reaches its maximum at a production level of 4,000 transistor radios per week, producing a maximum profit of 21,000 Br.

💡 Memory Hook

Sinking fund builds a future sum; amortization retires a present debt

📊 Synthesis Tables

Methods for Linear Systems

Method	Requirement	Solution cases
Inverse method	Square invertible coefficient matrix	Produces the solution vector
Gaussian elimination	Row reduction of the augmented matrix	Distinguishes unique, none, and infinitely many solutions

Types of Interest

Type	Interest basis	Typical use or feature
Simple interest	Initial principal only	Generally used for short-term loans or investments
Compound interest	Principal plus accrued interest	Generally used for long-term borrowing
Continuous compounding	Continuous accumulation represented by $e^{rt}$	Limiting case as conversion frequency increases