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Matrices and Determinants - Coggle Diagram
Matrices and Determinants
A matrix is a rectangular table of numbers or elements organized into rows and columns. It is used to represent and manipulate data in mathematics, physics, engineering, and computer science.
Formally:
A matrix A of size m × n is composed of elements aᵢⱼ, where:
i → row (1 to m)
Denoted: A = [aᵢⱼ]
Example:
A =
[2 1 3]
[4 -1 0]
→ 2 × 3 matrix
Main uses
Solve systems of linear equations
Represent linear transformations
Operate data (addition, subtraction, multiplication, inversion)
Applications in data science and programming
Order of an matrix
The order is defined by the number of rows (m) and the number of columns (n). It is expressed as m × n.
Example:
A =
[1 2 3]
[4 5 6]
→ order 2 × 3
Importance
Determines whether operations can be performed between matrices.
Addition/Subtraction: only if they have the same order.
Multiplication: possible only if columns of the first = rows of the second.
Dependent properties: trace and determinant (only for square matrices).
Special matrix types
Identity matrix
Square, diagonal with 1, the other elements 0.
Property: AI = IA = A
Example:
I₃ =
[1 0 0]
[0 1 0]
[0 0 1]
Triangular matrix
Upper: zeros below the diagonal.
Lower: zeros above the diagonal.
Useful in linear and determinant systems.
Symmetric matrix
A = Aᵀ holds (equal to its transpose).
Used in linear algebra, calculus, and physics.
Orthogonal matrix
Orthonormal columns.
Q⁻¹ = Qᵀ holds.
Applications: geometry, linear transformations.
Matrix Operations
Addition and subtraction
Only if they have the same order.
The corresponding elements are added/subtracted.
Properties: commutative, associative, distributive.
Example:
A=
[1 2 3]
[4 5 6],
B=
[7 8 9]
[10 11 12],
A+B=
[8 10 12]
[14 16 18]
Multiplication
Only if columns of A = rows of B.
Product AB → matrix of order m × p
Non-commutative: AB ≠ BA
Rule:
cᵢⱼ = Σ (aᵢₖ × bₖⱼ)
Example:
A (2×3) × B (3×2) → C (2×2)
C =
[58 64]
[139 154]
Multiplication by a scalar
Multiplies each element of the matrix by a number k.
Result has the same order as A.
Transposition
Convert rows to columns.
If A is of order m × n, Aᵀ will be n × m.
If A is symmetric, then A = Aᵀ.
Inverse matrix
Only for square and invertible matrices (det ≠ 0).
There exists a matrix A⁻¹ such that: AA⁻¹ = A⁻¹A = I.
Used to solve systems: Ax = b → x = A⁻¹b.
Not all matrices are invertible (singular matrices if det = 0).
Method: Gaussian elimination.
Elementary operations
They are applied to rows or columns to simplify matrices. These are used in the Gaussian elimination method.
Basic Operations
Swap two rows (or columns).
Multiply a row by a non-zero constant.
Add to one row another multiplied by a constant.
Properties
They do not alter the system's solution set.
They facilitate resolution by escalation.
Row reduction
Process for transforming a matrix into row echelon form.
Objective: To obtain zeros below the main diagonal.
Used to solve systems of linear equations.
Method
Choose pivot (first non-null element).
Making zeros below the pivot with elementary operations.
Repeat by columns until you get a triangular shape.
Solve with back substitution.
Example:
System
x + 2y – z = 4
2x – y + z = –1
3x + y + 2z = 6
→ Row echelon form:
[1 2 -1 | 4]
[0 5 -3 | 9]
[0 0 2 | 15]
→ Solution:
z = 7.5, y = 3, x = –2