Understanding Matrix Multiplication

Matrix Multiplication Definition

• Given matrices $A = (a_{ij}) \in K^{m \times n}$ and $B = (b_{jk}) \in K^{n \times p}$, their product $C = A \cdot B = (c_{ik}) \in K^{m \times p}$.
• The element $c_{ik}$ is calculated as $c_{ik} = \sum_{j=1}^{n} a_{ij} \cdot b_{jk}$ for $i = 1, \dots, m$ and $k = 1, \dots, p$.

Matrix Multiplication Example

• For matrices $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}$.
• Their product is $A \cdot B = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$.

Matrix Multiplication Properties

• Associativity: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
• Distributivity: $A \cdot (B + C) = A \cdot B + A \cdot C$ and $(A + B) \cdot C = A \cdot C + B \cdot C$
• Non-commutativity: Generally, $A \cdot B \neq B \cdot A$

Matrix Multiplication Specifics

• Multiplication is defined only if the number of columns in A equals the number of rows in B.
• The resulting matrix $A \cdot B$ has the same number of rows as A and the same number of columns as B.

Positive Economics

• Deals with factual statements that can be either true or false.

Normative Economics

• Deals with statements about what should be.
• Based on value judgments and cannot be factually proven true or false.

Economic Theories

• Theories explain how things work based on definitions, assumptions, and implications.

Assumptions in Economic Theories

• Simplifying assumptions reduce model complexity.
• Critical assumptions affect the theory's implications.

Testing Economic Theories

• Theories are tested by comparing their implications to empirical evidence.
• Depending on the results, theories may be abandoned, revised, or considered useful.

Economic Data: Index Numbers

• Index numbers express data relative to a base period using the formula: $\frac{absolute,value,in,given,period}{absolute,value,in,base,period} \cdot 100$.

Types of Economic Data

• Time-series data tracks a single variable over time.
• Cross-sectional data examines different variables at the same time.
• Scatter data plots one variable against another.

Positive Relationship in Economic Data

• As one variable increases, the other variable also increases.

Negative Relationship in Economic Data

• As one variable increases, the other variable decreases.

Linear Relationship in Economic Data

• Expressed as $y = a + bx$, where:
  • Slope: $b = \frac{\Delta y}{\Delta x}$.
  • Vertical intercept: $a$.

Nonlinear Relationship in Economic Data

• The slope changes as x changes.

Graphing Functions in Economics

• Plot the independent variable on the x-axis and the dependent variable on the y-axis for a function $y = f(x)$.

Examples of Functions in Economics

• $y = 5 + 2x$ is a linear function with a positive slope.
• $y = 5 - 2x$ is a linear function with a negative slope.
• $y = 5 + 2x - x^2$ is a non-linear function.

Slope of a Curve

• The slope is calculated as the slope of a straight line tangent to the curve at a given point.

Area Under a Curve

• Approximated by dividing the area into rectangles and summing their areas.
• Accuracy improves as the width of the rectangles approaches zero.