Matrices and Determinants Introduction

A matrix is a rectangular arrangement of numbers, symbols, or expressions, arranged in rows and columns. It is widely used in various branches of mathematics, including algebra, calculus, and statistics, as well as in applied fields like physics and computer science. Matrices are usually denoted by capital letters and enclosed within square brackets or parentheses.

Example of a Matrix:

Matrix (A) with 2 rows and 3 columns (2x3 matrix):

[ A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} ]

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. Determinants have important properties and applications in linear algebra, especially for solving systems of linear equations, finding inverses of matrices, and determining whether a matrix is invertible. The determinant is denoted as det(A) or (\lvert A \rvert).

Example of a Determinant:

For a 2x2 matrix (A):

[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} ]

The determinant of (A) is calculated as:

[ \text{det}(A) = ad - bc ]