Further Pure 1

1. Further Pure 1 Inverse Matrices

2. Reminder from lesson 1 • Note that any matrix multiplied by the identity matrix is itself. • And any matrix multiplied by the zero matrix is the zero matrix.

3. Inverse Matrix • All operations have an opposite. • We discussed in lesson 2 about using matrices to perform transformations. • An inverse matrix will undo the transformation and return you to where you started. • If a matrix is called A, then its inverse is known as A-1. • In lesson 1 we briefly met the concept of an identity matrix (seen on first slide). • So if multiplying A by A-1 returns you to where you started and multiplying by the identity matrix leaves you where you are, we can conclude that AA-1 = A-1A = I

4. Challenge 1 • You need to know the general formula for the inverse of a 2 × 2 matrix. • Can you find the inverse to the following matrix? • Use the property TT-1 = I • From this we can form 2 pairs of simultaneous equations. 2p + 3q = 1 2r + 3s = 0 p + 4q = 0 r + 4s = 1

5. Challenge 1 • The solutions to these equations are • So the inverse of T is

6. Challenge 1 • Lets now take any 2 × 2 matrix. • Can you use T and T-1 to find the inverse of M and hence the general formula for the inverse of any 2 × 2 matrix?

7. The Determinant of a 2 × 2 matrix • We have just found the general equation for the inverse of any 2 × 2 matrix. • The Δ symbol is a capital delta and will always be a numerical value. • The value can be calculated from the matrix and is known as the determinant of the matrix. • Using T and T-1 can you spot how to calculate it?

8. The Determinant of a 2 × 2 matrix • To calculate the determinant of a matrix M you multiply a by d and subtract b by c. • Below is the official notation. • If the det is zero then the inverse does not exist and the matrix is known as singular. • If the det is not zero then the inverse does exist and the matrix is known as non-singular. • Note: Only square matrices have inverses.

9. Challenge 2 • This is above the scope of the course and not required for you to do. • However it is a challenging question that will test your algebraic manipulation skills. • Can you find the inverse of M using the identity below and the method we used a few slides ago. • This will also prove where the formula for the determinant comes from.

10. Questions • Find the inverse of the following matrices.

11. Inverse of a product • Find the inverse of AB. • Lets call the inverse of AB, X. • So as we already know X(AB) = I • First multiply by B-1X(AB)B-1 = I×B-1 XA = B-1 • Next multiply by A-1XAA-1 = B-1A-1 X = B-1A-1 • This is an important result that you need to know (AB)-1 = B-1A-1

12. The orange square is an enlargement of the black square by a scale factor 2. What is the area of the object? Area = 9 units2 What is the area of the image? Area = 36 units2 The transformation performed can be described by the following matrix What do you notice about the determinant of the matrix and the enlargement shown. The determinant of a matrix indicates the scale factor of the area of enlargement. The det T is known as the signed scale factor as it can be negative. The negative signifies that the rotation direction has been reversed. Properties of the determinant

13. Task • Can you explain how we know that the area of any shape rotated θ degrees anti-clockwise about the origin remains the same.

14. Matrices with det = 0 • The determinant of a matrix tells us the scale factor of the areas` enlargement. • What would be the area of a shape transformed by a matrix with det = 0? • The area would be 0. • All the points will have been transformed so what will the image look like? • The image will be a straight line. • We can see an example of this on the next slide.

15. Lets start with a rectangle on a 2D pair of axes. We can write the co-ordinates of the vertices in matrix form. Next transform the object using a matrix with a det = 0 The image becomes a series of points that are in a straight line. Example 1

16. In fact although we used a rectangle for the example any point in the plane will transform to the line. From the diagram its clear to see what the equation of the line will be. y = x Example 1

17. We can reach the same result as the last slide using an algebraic method. Lets look at the general co-ordinate (x,y). Under the transformation we get the co-ordinates (x`,y`) Using matrix multiplication we can see that. x + 2y = x` x + 2y = y` From this we get y` = x + 2y = x` Or y = x Example 1

18. Example 2 • The plane is transformed by the matrix. • Show that the whole plane is mapped to a straight line and find the equation of this line. • Using matrix multiplication gives us the simultaneous equations. x` = 2x – y y` = -4x + 2y • From the equations we get y` = -2(2x – y) = -2x` • All the points will map to the line y = -2x

19. Example 2 • All points in a plane transform to a straight line. • This is because there are infinitely many lines that transform to a single point.

20. Example 3 • For the matrix T find the equation of the line of points that map to (5,-10). • We use matrix multiplication to find what equations will be equal to the co-ordinate (5,-10) • This gives us the equations 2x – y = 5 -4x + y = -10 • These two equations give the exact same information. 2x – y = 5

21. Summary 1 • The inverse of a matrix • Is • Where

22. Summary 2 • MM-1 = M-1M = I • X = B-1A-1