eJournal 6: Matrices & Transformations

Matrix Operations

– Addition & Subtraction
For adding/subtracting matrices of the same order, all you have to do is add/subtract normally with the corresponding elements in each matrix.

– Multiplication by a number
As for multiplying a matrix by a number, you have to multiply each element of the matrix with that particular number.

– Multiplication by another matrix
Matrices can only be multiplied by each other if they are compatible. This means that the number of columns in the left-hand matrix must be equal to the number of rows in the right-hand matrix. The inner part of the order of matrix must be the same, where the order of matrix is defined as the number of rows x number of columns. Another tip is that matrix multiplication isn’t commutative therefore, the product of AB doesn’t always equal to the product of BA.

The letters above show how matrices can be multiplied with each other. You have to multiply the row by the column where you should start by multiplying the first element of the first row with the first element of the first column, and so on.

The Inverse of a Matrix

Only square matrices possess an inverse.
I is known as the identity matrix where:
for each labelled matrix, I is equal to…

To find the inverse of a matrix, the formula is displayed below.

(ad – cb) is the determinant of the matrix and if the value of the determinant is equal to 0, that means the matrix has no inverse.

NOTE: multiplying by the inverse of a matrix has the same result as dividing by the matrix. This can be said in relation to ordinary algebraic operations.


Simple Transformations

• Reflection
  In reflections…
  – Every point is the same distance away from the mirror line
  – The reflection has the same size as the original image
  

• Rotation
  To conduct & fully describe a rotation, three things are needed:
  – the angle
  – direction
  – centre of rotation
  
  The object rotated is of the same shape and size but is rotated in different directions (clockwise/anticlockwise)
  

An easy method of conducting rotations is by using tracing paper. All you have to do is place the tip of a marker on the centre of rotation, making sure not to move it while rotating. Next, follow the instructions by rotating the object in the given direction and angle, using each quadrant as an indication for the angles.

• Translation
  Translation simply means to move the shape, without rotating, resizing, etc. Translation can be described using a column vector displayed in this form:

This means that the image moves 23 squares to the right and 7 squares up the page. The number at the top determines whether the object will move left or right, where moving to the left is displayed by a negative number and moving to the right is shown using a positive number. The number at the bottom determines whether the object will move up or down, where moving up is displayed using a positive number and moving down is shown using a negative number.

• Enlargement
  To describe an enlargement, the scale factor and the centre of enlargement are both required.

All you have to do is draw straight lines from the centre of enlargement to the corresponding vertices of the shape. Then, using this, you can find the coordinates of the enlarged shape using the scale factor. For example, if the scale factor is 3 and the line AC of the object took up only one square, you’ll know that the new line AC in the image will take up 3 squares instead, because 1 x 3 = 3. Given below is another example.

Transformations Using Matrices

Describing a Transformation Using Base Vectors

Real-Life Applications of Matrices and Transformations

1. Matrices can be used in physics-related applications, especially in the study of things like electrical circuits, quantum mechanics and optics. They mainly play a vital role in calculations of battery power outputs and resistor conversion of electrical energy into another useful form of energy.

2. Transformations are often used by architects. They can create a beautiful and appealing pattern by rotating, translating, flipping shapes, etc.

3. Another way that matrices are used in real life applications are in computer-based applications. They help in the projection of three dimensional images into a two dimensional screen, to create realistic and lively motions. Another useful computer-based application is that matrices help in the encryption of message codes.


IB Learner Profile

The first thing that matrices & transformations has taught me in terms of IB is to be knowledgeable. I’ve learned a lot from this topic, as well as how useful matrices and transformations can be in our real life in various fields of study. I’ve developed new skills and a much more thorough understanding as we studied this topic in great depth. Next, it has taught me to be a thinker, as in particular instances, we were told to use our creative and critical thinking skills to explore new ideas, especially when it came to the inverse of a matrix. This resulted in us gaining a lot of knowledge from the ideas, connections and theories we formed.