Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Algebra → Linear Algebra ↓


Matrices


Matrix is a fundamental concept in the field of mathematics, especially in linear algebra. Let us learn in detail about the concept of matrix, their types, operations, and diverse applications.

Introduction to matrices

A matrix is a two-dimensional array of numbers, arranged in rows and columns. The numbers inside a matrix are called its elements. Matrices can be used to solve systems of linear equations, among other applications.

For example, consider the following matrix:

A = | 1 2 3 | | 4 5 6 |A = | 1 2 3 | | 4 5 6 |

This matrix A has two rows and three columns. Therefore, it is a 2x3 (read as "two by three") matrix.

Visual example of a matrix

123456

Types of matrices

Matrices can be classified into different types depending on their size and the nature of their elements. Here are some common types:

Row matrix

A row matrix has only one row. For example:

B = | 1 2 3 |B = | 1 2 3 |

Column matrix

A column matrix has only one column. For example:

C = | 1 | | 2 | | 3 |C = | 1 | | 2 | | 3 |

Square matrix

A square matrix has the same number of rows and columns. For example:

D = | 1 2 | | 3 4 |D = | 1 2 | | 3 4 |

Diagonal matrix

A diagonal matrix is a square matrix where all diagonal elements are zero. For example:

E = | 1 0 | | 0 4 |E = | 1 0 | | 0 4 |

Matrix operations

Matrix operations are similar to arithmetic operations on numbers. We usually perform addition, subtraction, and multiplication on matrices.

Matrix addition

To add two matrices, their dimensions must be the same. Corresponding elements are added together. Consider matrices F and G:

F = | 1 2 | G = | 3 4 | | 5 6 | | 7 8 | F + G = | 1+3 2+4 | | 5+7 6+8 | = | 4 6 | | 12 14 |F = | 1 2 | G = | 3 4 | | 5 6 | | 7 8 | F + G = | 1+3 2+4 | | 5+7 6+8 | = | 4 6 | | 12 14 |

Matrix subtraction

Matrix subtraction is similar to addition, where we subtract corresponding elements:

F - G = | 1-3 2-4 | | 5-7 6-8 | = | -2 -2 | | -2 -2 |F - G = | 1-3 2-4 | | 5-7 6-8 | = | -2 -2 | | -2 -2 |

Matrix multiplication

Matrix multiplication is more complicated than addition or subtraction. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

For example, if H is a 2x3 matrix and I is a 3x2 matrix, they can be multiplied:

H = | 1 2 3 | I = | 1 2 | | 4 5 6 | | 3 4 | | 5 6 | HI = | (1×1 + 2×3 + 3×5) (1×2 + 2×4 + 3×6) | | (4×1 + 5×3 + 6×5) (4×2 + 5×4 + 6×6) | = | 22 28 | | 49 64 |H = | 1 2 3 | I = | 1 2 | | 4 5 6 | | 3 4 | | 5 6 | HI = | (1×1 + 2×3 + 3×5) (1×2 + 2×4 + 3×6) | | (4×1 + 5×3 + 6×5) (4×2 + 5×4 + 6×6) | = | 22 28 | | 49 64 |

Transposition of a matrix

The transpose of a matrix is obtained by inverting the rows and columns. For example, the transpose of a matrix A is written as A T:

A = | 1 2 3 | A T = | 1 4 | | 4 5 6 | | 2 5 | | 3 6 |A = | 1 2 3 | A T = | 1 4 | | 4 5 6 | | 2 5 | | 3 6 |

Determinants and inverses

Determinants and inverses are important in solving systems of linear equations and in various matrix applications.

Determinants

The determinant is a scalar value that can be calculated from the elements of a square matrix and it encodes some properties of the matrix.

For a 2x2 matrix J:

J = | ab | | cd | det(J) = ad - bcJ = | ab | | cd | det(J) = ad - bc

Inverse

The inverse of a matrix K, denoted by K - 1, is a matrix that, when multiplied by K, gives the identity matrix.

A matrix has an inverse only if its determinant is not zero. For a 2x2 matrix:

K = | ab | | cd | K -1 = (1/det(K)) * | d -b | | -ca | det(K) = ad - bc (should not be zero)K = | ab | | cd | K -1 = (1/det(K)) * | d -b | | -ca | det(K) = ad - bc (should not be zero)

Applications of matrices

Matrices are used in various fields such as computer graphics, engineering, physics, and statistics. They are essential in representing transformations and in modeling complex systems.

Computer graphics

In computer graphics, matrices represent transformations such as rotation, scaling, and translation applied to images and 3D models.

Systems of linear equations

Matrices are used to solve systems of linear equations. They simplify complex calculations and are used in algorithms to solve equations efficiently.

Economics and statistics

In economics matrices are used in input-output models and in statistics they are used in covariance matrices and linear regression models.

Conclusion

Understanding matrices is important in many areas of mathematics and its applications. By performing matrix operations and learning how to work with special types of matrices, we can solve real-world problems efficiently and effectively.


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