Comprehensive Study Notes for Exam Preparation
Matrices
Definition: Each matrix has a size, notation m \times n, where m represents the number of rows and n represents the number of columns.
Square Matrix: A matrix where the number of rows is equal to the number of columns.
Matrix Equality: Two matrices are equal if and only if all their corresponding entries are the same, and they have the same size.
Addition and Subtraction: Operable only when matrices have the same size.
Example: A - B = \begin{bmatrix} a & b \ c & d \end{bmatrix} - \begin{bmatrix} e & f \ g & h \end{bmatrix} = \begin{bmatrix} a-e & b-f \ c-g & d-h \end{bmatrix}
Scalar Multiplication: Each entry in the matrix is multiplied by the scalar number.
Example: k \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \ kc & kd \end{bmatrix}
Transpose (A^T):
If A is an m \times n matrix, its transpose A^T is an n \times m matrix. Rows become columns and columns become rows.
Example: If A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, then A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}
Properties of Transpose (page 3):
(A^T)^T = A
(kA)^T = kA^T (where k is a scalar)
(A+B)^T = A^T + B^T
Additive Inverse: The additive inverse of matrix A is -A.
Product of Two Matrices (AB):
Condition: The product AB is only possible if the inner dimensions of the matrices are the same. If A is m \times r and B is r \times n.
Resulting Size: The resulting matrix AB will have the size of the outer dimensions, i.e., m \times n.
Process of Multiplication: Each entry in the resulting matrix is found by taking the dot product of a row from the first matrix and a column from the second matrix.
Example: \begin{bmatrix} a & b \ c & d \end{bmatrix} \begin{bmatrix} e & f & i \ g & h & j \end{bmatrix} = \begin{bmatrix} ae+bg & af+bh & ai+bj \ ce+dg & cf+dh & ci+dj \end{bmatrix}
Important Note: Matrix multiplication is not commutative, meaning AB \neq BA in general.
Calculator Usage for Graphing (page 15)
Entering Matrices: Go to "2nd", click on "x^{-1}", select "edit", specify the size (m \times n), and fill in the entries.
Graphing Lines: Go to "Y=", enter your equation, then click "Graph". If the graph is not visible on the screen, click "Zoom" and then "Zstandard" to adjust the viewing window.
Plotting Points & Graphing Lines
Slope (m) between two points (x1, y1) and (x2, y2)
Definition: m = \frac{\text{Change in y}}{\text{Change in x}} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y2 - y1}{x2 - x1}
Special Cases:
If x1 = x2: The slope m is undefined. This represents a vertical line, and its equation is x = x_1.
If y1 = y2: The slope m = 0. This represents a horizontal line, and its equation is y = y_1.
Forms of Linear Equations (page 21):
Point-Slope form: y - y1 = m(x - x1)
Slope-Intercept form: y = mx + b (where m is the slope and b is the y-intercept).
Standard form: Ax + By = C (where A, B, and C are integers).
Intercepts:
x-intercept: The point where the graph crosses the x-axis ((#, 0)). To find it, set y = 0 in the equation.
y-intercept: The point where the graph crosses the y-axis ((0, #)). To find it, set x = 0 in the equation.
Linear Depreciation (page 26)
Definition: A linear mathematical model used to represent how the value of an asset decreases over time.
Points: Represented as (t, V(t)), where t is time and V(t) is the value of the asset at time t.
Model Formula: V(t) = mt + b
t: time.
V(t): value of the asset at time t.
b: initial value of the asset, equivalent to V(0).
m: constant rate of change of the asset's value. For depreciation, m is negative (m < 0).
Rate of Depreciation: The absolute value of the slope, |m|. When stating the rate, the negative sign is typically omitted as 'depreciation' already implies a decrease.
Scrap Value: The lowest value an asset can attain, which may not necessarily be zero. If scrap value is mentioned, it's crucial to check if the linear model applies throughout the asset's life or only up to the scrap value (refer to examples on page 27).
Cost, Revenue, and Profit
Total Cost (C(x)):
Definition: The sum of variable costs and fixed costs incurred in production.
Formula: C(x) = \text{Variable Costs} + \text{Fixed Costs} = (\text{Production cost per item}) \times (\text{quantity}) + \text{Fixed Costs}
Linear Model: C(x) = mx + F
x: quantity (number of items produced).
F: Fixed Costs, which are constant regardless of production quantity (y-intercept (0, F)).
m = \frac{\Delta C}{\Delta x}: Production Cost Per item (slope).
Note: Always remember to include fixed costs (F) when defining the total cost function.
Revenue (R(x)):
Definition: The money a company generates from sales of its products.
Formula: R(x) = (\text{Price per item}) \times (\text{quantity})
Linear Model: R(x) = P \cdot x
x: quantity (number of items sold).
P: Price per item (assumed constant in this model).
Points: (x, R(x)), with the graph typically passing through the origin (0,0).
Profit (P(x)):
Definition: The amount of money a company retains after deducting all expenses from its revenue.
Formula: P(x) = R(x) - C(x) (Revenue minus Cost).
Note: It is crucial to enclose the entire cost function C(x) in parentheses when subtracting to ensure the negative sign is distributed to all terms within C(x) (e.g., P(x) = R(x) - (mx + F)). The y-intercept of the profit function is (0, -F).
Notation: "p" (lowercase) is used for price, and "P" (uppercase) is used for profit.
Break-Even Point (x0, R0 or x0, C0):
Definition: The level of sales (quantity) at which a company's total revenues equal its total costs, resulting in zero profit (P(x) = 0).
It represents the minimum sales quantity needed for the company to begin making a profit.
Steps to find the Break-Even Point:
Set R(x) = C(x) or P(x) = 0 and solve for x. This value of x is the break-even quantity (x_0).
Plug the break-even quantity (x0) into either the revenue function (R(x0)) or the cost function (C(x0)) to find the break-even revenue (R0).
State your answer and explanation: "It costs (C0) for the company to produce (x0) items, but when they are sold, the same amount of revenue (R_0) is brought in."
Demand and Supply
Demand (P(x)):
Definition: The relationship between the price of a good or service and the quantity consumers are willing and able to purchase at that price.
Behavior: As the number of items a consumer needs (quantity demanded) increases, the price they are willing to pay typically decreases.
Linear Model: P(x) = mx + b
P: Price.
x: Number of items purchased (quantity).
Slope: The slope (m) of a demand curve is typically negative (m < 0).
Points: (x, P).
Supply (P(x)):
Definition: The relationship between the price of a good or service and the quantity producers are willing and able to offer for sale.
Behavior: The more items a producer has to sell, the higher the price they want to sell them for.
Linear Model: P(x) = mx + b
P: Price.
x: Number of items supplied (quantity).
Slope: The slope (m) of a supply curve is typically positive (m > 0).
Points: (x, P).
Equilibrium Point (x0, P0):
Definition: The point where the Supply and Demand curves intersect. At this point, the quantity demanded by consumers equals the quantity supplied by producers.
Steps to find the Equilibrium Point:
Set the Supply function equal to the Demand function (S(x) = D(x)) and solve for x. This value is the equilibrium quantity (x_0).
Plug the equilibrium quantity (x0) into either the Supply function (S(x0)) or the Demand function (D(x0)) to find the equilibrium price (P0).
State your answer and explanation: "At a price of (P0) per item, producers will supply (x0) items, and at the same price, consumers will buy (x_0) items."
Systems of Linear Equations (page 39)
Independent System:
Solution: Has exactly one unique solution (x,y).
Graphical Representation: The lines intersect at a single point.
Slopes: The lines have different slopes (m1 \neq m2).
Inconsistent System:
Solution: Has no solutions.
Graphical Representation: The lines are parallel and never intersect.
Solving Outcome: While solving algebraically, you will arrive at a contradiction (an untrue statement like 0 = 5).
Slopes & Intercepts: The lines have the same slope (m1 = m2) but different y-intercepts (b1 \neq b2).
Dependent System:
Solution: Has infinitely many solutions.
Graphical Representation: The lines overlap; they are the same line.
Solving Outcome: While solving algebraically, you will arrive at an always true statement (an identity like 1 = 1).
Slopes & Intercepts: The lines have the same slopes (m1 = m2) and the same y-intercepts (b1 = b2).
Solution Description: Requires a parametric solution to express all points on the line.
Parametric Solution (for Dependent Systems):
Steps:
Choose one variable as the "free variable" (e.g., x).
Introduce a parameter (e.g., t).
Let the free variable be equal to the parameter (e.g., let x = t).
Rewrite the other variable(s) in terms of the chosen parameter.
Write the solution as an ordered pair (or triplet for 3 variables) expressed in terms of the parameter (e.g., (x, y) = (t, \text{expression in } t)).