## Matrices

A matrix (plural: matrices) is an array of numbers arranged in rows and columns and enclosed in brackets.
Matrices are used, at this level, to represent vectors and transformations and can be used to solve simultaneous equations

### Definitions

• A number in a matrix is called an element.
• row of a matrix is horizontal.
• column of a matrix is vertical.
• The order of a matrix is the number of rows followed by the number of columns.

e.g. is a 2 X 3 matrix (read as 2 by 3). It has 2 rows and 3 columns.

• Equal matrices have the same order and their corresponding elements are equal.

e.g. (3 4) = (3 4) but (3 4) ≠ (4 3)

• zero matrix has every element equal to zero.

e.g. (0 0) and  are zero matrices

• The determinant of a 2 X 2 matrix, written Δ or |M|, is the difference of the products of the two diagonals.

• singular matrix has a determinant of zero.

e.g. A =  |A| = 0 therefore A is a singular matrix.

### Addition and Subtraction of Matrices

Matrices can be added and subtracted only when the order of each matrix is the same.
The corresponding elements are then added or subtracted.

e.g.

cannot be added as they are of different orders.

### Multiplication of Matrices

• A matrix can be multiplied by an ordinary number (called a constant or scalar). Each element of the matrix is multiplied by the number.

e.g.

• Two matrices can be multiplied together ONLY if they are compatible. To be compatible, the number of columns of the first matrix has to be the same as the number of rows of the second matrix. The elements of each row of the first matrix are then multiplied by the elements of each column of the second matrix.

Example 1

1 X 3 matrix multiplied by a 3 X 1 matrix gives a 1 X 1 matrix.

Example 2

2 X 2 matrix multiplied by a 2 1 matrix gives a 2 X 1 matrix.