1. Definition & Types of Matrices

A matrix is a rectangular array of numbers or expressions arranged in rows and columns. A matrix of order m × n has m rows and n columns.

• Row matrix: one row.
• Column matrix: one column.
• Square matrix: m = n.
• Rectangular matrix: m ≠ n.
• Zero matrix: all entries are 0.
• Diagonal matrix: non-zero entries only on main diagonal.
• Scalar matrix: diagonal matrix where all diagonal entries are equal.
• Identity matrix (I): diagonal entries = 1.
• Symmetric matrix: A = A^T.
• Skew-symmetric matrix: A^T = −A (diagonal entries are 0).

2. Equality of Matrices

Two matrices A and B are equal if they have the same order and their corresponding elements are equal: A = B ⇔ aij = bij for all i, j.

3. Operations on Matrices

(A + B)ij = aij + bij

(kA)ij = k · aij

Matrix Multiplication

Product AB is defined when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p.

(AB)ij = \sum_{k=1}^{n} aik bkj

Note: Matrix multiplication is not commutative in general: AB != BA.

4. Properties of Addition & Scalar Multiplication

• Commutative: A + B = B + A
• Associative: (A + B) + C = A + (B + C)
• Additive identity: A + O = A
• Additive inverse: A + (−A) = O
• Distributive: k(A + B) = kA + kB

5. Properties of Matrix Multiplication

• Associative: A(BC) = (AB)C
• Distributive: A(B + C) = AB + AC and (B + C)A = BA + CA
• Identity: AI = IA = A
• Not commutative in general: AB != BA
• Zero product does not imply a zero factor: AB = O doesn't force A = O or B = O

6. Transpose of a Matrix

The transpose AT is obtained by interchanging rows and columns: the element at row i, column j of A becomes row j, column i of AT.

• (AT)T = A
• (A + B)T = AT + BT
• (kA)T = kAT
• (AB)T = BT AT

7. Symmetric and Skew-Symmetric Matrices

Symmetric: A = A^T.
Skew-symmetric: A^T = −A (diagonal entries are 0).

Any square matrix A can be uniquely decomposed as:

A = (A + AT)/2 + (A − AT)/2

The first term is symmetric; the second term is skew-symmetric.

8. Special Results & Useful Facts

• Product of two symmetric matrices is symmetric only if they commute: if AB = BA, then (AB)T = AB.
• A AT is always symmetric.
• Trace of a matrix (sum of diagonal elements) is linear: tr(A + B) = tr(A) + tr(B) and tr(kA) = k·tr(A).

9. Applications

Matrices are widely used for:

• Solving systems of linear equations (precursor to determinants and Cramer’s rule)
• Linear transformations and geometry
• Computer graphics and image transforms
• Economics, statistics and data modelling
• Physics (e.g. tensors and quantum mechanics)