Overview

 

• Matrix Addition and Subtraction  

• Matrix Multiplication  

• Matrix Transformations 

 

 

Matrix Addition and Subtraction  

 

How to add matrices together: 

 

• In order to add matrices together they must have the same dimensions  

• For example:

​

​

​

​

​

• There is no solution as the two matrices have different dimensions 

• However, when they have a same dimensions addition and subtraction is possible 

• In order to add matrices we just add the numbers in each position together 

• For example: 

 

​

​

 

 

Practice Question: 

 

1. 

 

​

​

​

2.

​

 

 

 

 

Solutions : 

 

​1.​

 

 

​

​

2.

 

​

​

​

​

 

Matrix Multiplication  

 

 How to know if matrix multiplication is possible: 

 

• We need to look at the dimensions of both matrices 

• If the number of columns on one matrix is equal to the rows of the other then multiplication is possible: 

 

​

​

​

​

• Here our first matrix is a 2x2 and the second matrix is a 2x1 

• As the number of columns on the first matrix is 2 and the number of rows on the second matrix is 2, then multiplication is possible 

 

How to find what the dimensions the new matrix is: 

 

• To find out the dimensions of the new matrix we need to look at the rows of the first matrix and the columns of the second: 

 

​

​

​

​

• Here the first matrix has 2 rows and the second matrix has 1 column 

• Therefore the matrix formed is a 2x1 matrix 

 

How to multiply matrices: 

 

• Once we know the dimensions of the new matrix we have consider each number in the matrix 

• For example: 

​

​

​

​

​

• For a we need to multiply the top row of the first matrix by the column in the second matrix 

 

 

​

• For be we need to do the same but with the bottom row of the first matrix 

​

​

​

• Therefore the answer is 

 

​

 

 

 

Practice Questions: 

​

​

1. 

 

 

​

 

2.

​

​

​

Solutions: 

 ​

1.

 

 

 

 

​

​

​

​

​

​

2.

​

​

​

​

​

​

​

​

​

​

​

 

 

Matrix Transformations 

 

 

The identity matrix: 

​

​

​

​

​

​

• Multiplying by it has no effect 

 

 

​

​

​

​

Different Types of transformation matrices: 

 

• Rotation 90 degrees anticlockwise around origin: 

 

 

​

​

​

• Rotation 180 degrees anticlockwise around origin: 

 

 

 

 

 

• Rotation 270 degrees anticlockwise around origin: 

 

 

 

 

 

• Reflections in line x = 0: 

 

 

​

 

 

• Reflection in line y = 0: 

​

​

​

​

 

• Reflection in line y = x: 

​

​

​

​

​

​

• Reflection in line y = -x: 

 

​

​

​

​

 

 

 

Combining Matrices: 

 

• If a point is transformed by matrix A followed by matrix B, we are able to combine these transformations to be represented by 1 matrix 

• This matrix is equal to BA  

• For example:

​

​

 

 

​

​

​

​

​

​

 

• Here we must do B x A in order to find C 

 

​

​

 

​

​

​

​

• Hence this is the answer 

​

​

​

​

 

Practice Question: 

​

​

​

​

​

​

​

​

​

​

​

 

​

​

Solution: