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Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
Probability & Statistics
Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation

Matrix Operations Visualization

Operation:
+
=

Matrix Basics

I. Definition of Matrix

A matrix is a rectangular array of numbers. An m×n matrix consists of m rows and n columns of elements arranged in a rectangular format.

General form:

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

Where \(a_{ij}\) represents the element in the i-th row and j-th column.

II. Basic Matrix Operations

1. Matrix Addition

Two matrices of the same type (same number of rows and columns) can be added by adding corresponding elements:

$$A + B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}$$

2. Matrix Subtraction

Similar to addition, subtract corresponding elements:

$$A - B = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} - \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11}-b_{11} & a_{12}-b_{12} \\ a_{21}-b_{21} & a_{22}-b_{22} \end{bmatrix}$$

3. Scalar Multiplication

Multiply a matrix by a scalar: each element of the matrix is multiplied by that scalar:

$$k \cdot A = k \cdot \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} k \cdot a_{11} & k \cdot a_{12} \\ k \cdot a_{21} & k \cdot a_{22} \end{bmatrix}$$

III. Properties of Matrix Operations

  • Commutative Law: \(A + B = B + A\)
  • Associative Law: \((A + B) + C = A + (B + C)\)
  • Distributive Law: \(k(A + B) = kA + kB\)
  • Zero Matrix: \(A + O = A\) (where O is the zero matrix)

IV. Examples

Let matrix \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\), \(B = \begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix}\), and scalar \(k = 2\)

Addition: \(A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}\)

Subtraction: \(A - B = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix}\)

Scalar Multiplication: \(2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}\)