Intro to Matrices

Matrices

• A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns.
• Matrices are widely used in mathematics, physics, computer science, and other fields.

Matrix Example

• The matrix A with m rows and n columns looks like this:

$\qquad A = \begin{bmatrix} \qquad a_{11} & a_{12} & \cdots & a_{1n} \
\qquad a_{21} & a_{22} & \cdots & a_{2n} \
\qquad \vdots & \vdots & \ddots & \vdots \
\qquad a_{m1} & a_{m2} & \cdots & a_{mn} \qquad \end{bmatrix}$

• $a_{ij}$ represents the element in the i-th row and the j-th column

Dimension

• The dimension of a matrix is denoted as $m \times n$, where m is the number of rows and n is the number of columns

Types of Matrices

• Square matrix: The number of rows equals the number of columns ($m=n$)
• Zero matrix: All elements are equal to zero
• Identity matrix: A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros
• Row matrix: A matrix with only one row
• Column matrix: A matrix with only one column

Matrix Operations

Addition

• Two matrices A and B can be added if they have the same dimensions.
• Matrix C results if each element $c_{ij} = a_{ij} + b_{ij}$.
• $C = A + B$

Subtraction

• Subtraction works the same way as addition, but instead of adding the elements you subtract them
• $C = A - B$

Scalar Multiplication

• Matrix A can be multiplied by a scalar k by multiplying each element in the matrix by k.
• $B = kA$

Matrix Multiplication

• Matrix multiplication of two matrices A ($m \times n$) and B ($n \times p$) is possible if the number of columns in A is equal to the number of rows in B.
• The result is a matrix C with dimension m x p, where each element $c_{ij}$ is calculated as:
• $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$

Transposition

• The transpose of a matrix A, denoted $A^T$, is obtained by swapping rows for columns.
• If A has dimension m x n, then $A^T$ has dimension n x m.
• $(A^T){ij} = a{ji}$

Inverse Matrix

• The inverse of a square matrix A, denoted $A^{-1}$, is a matrix such that:
• $AA^{-1} = A^{-1}A = I$
• where I is the identity matrix.
• Not all matrices have an inverse.
• A matrix that has an inverse is called invertible or non-singular.

Determinant

• The determinant of a square matrix A, denoted det(A) or |A|, is a scalar value that can be calculated from the elements of the matrix.
• The determinant is used to determine whether a matrix is invertible and to solve linear equation systems.

Calculation of Determinant

• For a 2 x 2 matrix:

$\qquad A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$

• The determinant is:
• $det(A) = ad - bc$
• For larger matrices, the determinant can be calculated using various methods, including cofactor expansion and row reduction.

Linear Equation Systems

• Matrices are often used to represent and solve linear equation systems.
• A linear equation system can be written in matrix form as:
• $Ax = b$
• where A is the coefficient matrix, x is the column matrix of variables, and b is the column matrix of constants.

Solving Linear Equation Systems

• There are several methods to solve linear equation systems, including:
  • Gauss elimination: Converts matrix A into an upper triangular matrix through row operations.
  • Inverse matrix: If A is invertible, the solution can be obtained as $x = A^{-1}b$
  • Cramer's rule: Uses determinants to solve the system.

Introduction to Probabilities

• Probability is a measure of the likelihood of an event occurring.
• Expressed as a number between 0 and 1. A 0 represents impossibility, and a 1 is certainty.

Examples

• The probability of flipping a coin and getting heads is 0.5.
• The probability of rolling a 3 on a six-sided die is 1/6.
• The probability of rain tomorrow is 0.25.

Probability Calculation

• The probability of an event is calculated by dividing the number of favorable outcomes by the number of possible outcomes.

$$ \qquad P(A) = \frac{\text{Number of favorable outcomes to A}}{\text{Number of possible outcomes}} $$

Example

• What is the probability of rolling an even number on a six-sided die?
  • Favorable outcomes: 2, 4, 6 (3 outcomes)
  • Possible outcomes: 1, 2, 3, 4, 5, 6 (6 outcomes)
  • Probability: $P(\text{even}) = \frac{3}{6} = 0.5$

Types of Events

• Certain Event: An event that always occurs (Probability = 1)
• Impossible Event: An event that never occurs (Probability = 0)
• Simple Event: An event with only one outcome
• Compound Event: An event with more than one outcome
• Mutually Exclusive Events: Events that cannot occur at the same time
• Independent Events: Events that do not affect each other

Properties of Probability

• The probability of an event always lies between 0 and 1.
• The sum of the probabilities of all possible events is 1.
• The probability of the union of two mutually exclusive events is the sum of their probabilities
• The probability of the intersection of two independent events is the product of their probabilities.
  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
  • $P(A \cap B) = P(A) \cdot P(B)$ if A and B are independent.

Venn Diagram

• A Venn diagram displayed includes two sets A and B, with the shaded regions:
  • A: Represents event A.
  • B: Represents event B.
  • $A \cap B$: Represents the intersection of A and B (events that occur simultaneously).
  • $A \cup B$: Represents the union of A and B (at least on event occurs).
  • Sample space (S): Set of all possible outcomes.
• Venn diagrams are useful for visualizing relationships between events and calculating probabilities.