Introduction

Matrices and determinants play a fundamental role in linear algebra and have wide applications in various fields such as engineering, physics, economics, and computer science.

Matrices

Definition of a Matrix

A matrix is a rectangular array of numbers, arranged in rows and columns. A matrix with m rows and n columns is called an m × n matrix.

A = [ 1 2 3 ]
    [ 4 5 6 ]
    [ 7 8 9 ]

Types of Matrices

• Row Matrix: A matrix with only one row.
• Column Matrix: A matrix with only one column.
• Square Matrix: A matrix where the number of rows equals the number of columns.
• Diagonal Matrix: A square matrix where all non-diagonal elements are zero.
• Identity Matrix: A diagonal matrix where all diagonal elements are 1.

Operations on Matrices

Matrix Addition and Subtraction

If two matrices have the same order, their sum and difference are obtained by adding or subtracting corresponding elements.

Matrix Multiplication

If A is an m × n matrix and B is an n × p matrix, their product AB is an m × p matrix.

Determinants

Definition of a Determinant

A determinant is a scalar value associated with a square matrix, which helps determine the invertibility of a matrix.

Determinant of a 2 × 2 Matrix

|A| = ad - bc

Determinant of a 3 × 3 Matrix

|A| = a(ei - fh) - b(di - fg) + c(dh - eg)

Inverse of a Matrix

The inverse of a matrix A exists if |A| ≠ 0 and is given by:

A⁻¹ = (1/|A|) * adj(A)

Applications of Matrices and Determinants

• Solving simultaneous equations
• Computer graphics and image processing
• Cryptography
• Economics and business modeling