Further Mathematics

Matrices (Pure Mathematics)

# Matrices

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Matrices

**Introduction to Matrices**

- A
**matrix**is a rectangular arrangement of numbers. - These numbers are called the
**elements**or**entries**of the matrix. **Rows**run horizontally and**columns**run vertically in a matrix.- The
**dimension**of a matrix is given in the form 'm x n' (m rows by n columns). - A
**square matrix**has the same number of rows as it does columns. - An
**identity matrix**is a square matrix with ones along the main diagonal and zeroes elsewhere.

**Matrix Operations**

- Matrices can be
**added**or**subtracted**element by element if they are of the same dimensions. - Matrix
**multiplication**is possible if the number of columns in the first matrix equals the number of rows in the second. - In the multiplication of matrices, order matters (
**AB ≠ BA**). - The
**transpose**of a matrix is obtained by swapping the rows and columns. - A matrix can be
**multiplied by a scalar**; each element is multiplied by the scalar.

**Determinants and Inverses**

- The
**determinant**of a matrix is a specific number that can only be calculated for square matrices. - If the determinant of a matrix is zero, it is said to be
**singular**or**non-invertible**. - The
**inverse**of a matrix A is denoted as A^-1, and it satisfies the property that A*A^-1 = A^-1*A = I, where I is the identity matrix. - To find the inverse of a matrix, one has to use the formula A^-1 = 1/det(A) adj(A), where adj(A) is the adjugate of A.

**Applications of Matrices**

- Matrices are used to solve systems of linear equations through Gaussian elimination or Cramer's rule.
- They are used to perform linear transformations including rotation, dilation, reflection and shearing.
- In computer graphics, matrices are used to manipulate and transform 3D figures.
- In probability and statistics, matrices are used in Markov chains to model and predict behaviour over time.