Mastering Matrices: From Basics to Operations in Linear Algebra

1 Matrix and Vector

If m and n are positive integers, a matrix has the form of an m × n array where each entry is a number. A matrix has rows and columns and is usually denoted by an uppercase letter.

2 Special Matrices

Row vector

Column vector

Square matrix

Diagonal matrix

Identity matrix

Zero matrix

3 Matrix Addition and Scalar Multiplication

3.1 Addition

If A and B are matrices of the same dimensions, their sum is a matrix obtained by adding corresponding entries.

Only matrices with equal numbers of rows and columns can be added.

3.2 Scalar Multiplication

If A is a matrix and k is a scalar, the product kA is a matrix whose entries are each multiplied by k.

3.3 Properties of Addition and Scalar Multiplication

1. Commutative law

2. Associative law

3. Distributive law (matrix addition over scalar multiplication)

4. Distributive law (scalar multiplication over matrix addition)

5. ... (additional standard properties)

4 Matrix Operations

4.1 Matrix Multiplication

If A is an m × p matrix and B is a p × n matrix, their product AB is an m × n matrix defined by the dot‑product of rows of A with columns of B.

4.2 Properties of Matrix Multiplication

1. Associative property of matrix multiplication

2. Distributive property over matrix addition

3. Distributive property over scalar multiplication

4. Associative property with scalar multiplication

5 Matrix Transpose

In linear algebra, the transpose of a matrix is an operator that flips the matrix over its diagonal, swapping rows and columns to produce a new matrix.

5.1 Properties of the Transpose

(Standard properties such as (Aᵀ)ᵀ = A, (A + B)ᵀ = Aᵀ + Bᵀ, (AB)ᵀ = BᵀAᵀ, etc.)