SHOP ATVRADIO DAY 153 DAY 278 DAY 345 SHOP BTVRADIO DAY 194 DAY 285 DAY 363 TOTALTVRADIO DAY 1147 DAY DAY 3108 This can be written in matrix form as: The tables below show the sales of TV’s and radios over three days in shop A and shop B. These can be combined to give the total sales of TV’s and radios for shop A and shop B.

Two matrices can be added together if they are the same size. The size of a matrix is called the order of the matrix. Example 2 rows 4 columns Note: the number of rows is always written first.

Find Both matrices are 2 × 2 so they can be added. add corresponding terms Example 1

Find Both matrices are 3 × 2 so they can be subtracted. subtract corresponding terms Example 2

Multiplying a matrix by a number So 2 A is the same as multiplying each term in the matrix A by the number 2. Example

Multiplying a row matrix by a column matrix SHOP ATVRADIO DAY 153 DAY 278 DAY 345 If a TV costs $500 and a radio costs $40, you can work out the amount of money shop A takes for the TV’s and radios on day 1 as follows: You can write this in matrix form as: Similarly for day 2: Similarly for day 3:

Example multiply each number in the row matrix with each number in the column matrix 1

Example multiply each number in the row matrix with each number in the column matrix 2

Example multiply each number in the row matrix with each number in the column matrix 3

Multiplying matrices In the last section you saw that when a TV costs $500 and a radio costs $40 SHOP ATVRADIO DAY 153 DAY 278 DAY 345 These can be combined into just one matrix equation. order3 × 2 2 × 1 3 × 1 same

It is only possible to multiply two matrices when the number of columns in the first matrix is the same as the number of rows in the second matrix. In general: First matrixSecond matrix Order Youcan multiply the matrices if The product of the two matrices will be of order

Example 1 The numbers in the middle are the same so it is possible to multiply the matrices. You must multiply each row in the first matrix with the column in the second matrix to find the missing numbers.

Example 2 You must multiply each row in the first matrix with each column in the second matrix to find the missing numbers.

2 ✕ 2 matrices

The determinant of a 2 ✕ 2 matrix

STEP 1 multiply the numbers on the leading diagonal STEP 2 multiply the numbers on the other diagonal STEP 3 subtract Example 1

STEP 1 multiply the numbers on the leading diagonal STEP 2 multiply the numbers on the other diagonal STEP 3 subtract Example 2

Inverse matrices The inverse of matrix A is denoted by A −1. The inverse matrix is calculated using the following formula: If AB = I, where I is the identity matrix, then B is called the inverse of A.

STEP 1 Example 1 find the determinant STEP 2 swap the numbers on the leading diagonal change the signs of the numbers on the other diagonal STEP 3 divide by the determinant

STEP 1 Example 2 find the determinant STEP 2 swap the numbers on the leading diagonal change the signs of the numbers on the other diagonal STEP 3 divide by the determinant

STEP 1 Example 3 find the determinant STEP 2 swap the numbers on the leading diagonal change the signs of the numbers on the other diagonal STEP 3 divide by the determinant

A matrix has no inverse when the determinant is zero because you cannot divide by zero. A matrix with no inverse is called a singular matrix. A matrix with an inverse is called a non-singular matrix.