Matrices I Cheat Sheet AQA A Level Further Maths: Core Multiply the transformation matrices together in reverse order. Matrix Arithmetic and Multiplying a Matrix by a Scalar Zero and Identity Matrices A matrix is an array of numbers, or elements, arranged in rows and columns. Below are some examples of matrices: In zero matrices, all elements are zero. They can come in any order. Below are all examples of zero matrices: 7 4 2 π΄=[ ] 4 9 13 6 4 8 π΅=[ ] 10 5 1 4 2 πΆ = [7 4] 6 3 The order of a matrix is described by the number of rows and columns in the form π × π. In the above examples, the order of π΄ and π΅ are 2 × 3 and the order of πΆ = 3 × 2. [0 0 0] 7 4 6 4 2 0 ]+[ ]−[ ] 4 9 10 5 6 11 Subtract numbers in corresponding positions. 0 0 0 [ ] 0 0 0 1 0 πΌ=[ ] 0 1 The transformation of an object into an image can be represented using matrices. The transformation matrix 1 0 can be found by mapping the transformation of two unit vectors, π = [ ] and π = [ ]. The resulting position 0 1 vectors form the transformation matrix. For example, when reflected in the π₯ axis, the images of π and π have 1 0 1 0 the position vectors π = [ ] and π = [ ]. Thus, the matrix for reflection in the π₯ axis is [ ]. 0 0 −1 −1 The orders of matrices show whether the matrices are conformable for multiplication and what matrix order the final product will have. The number of columns in the first matrix must be equal to the number of rows in the second. For example: (2 × 1) × (1 × 3) • • The middle numbers are the same, showing that the matrices are conformable for multiplication. The first and last numbers show that the final product will have an order of 2 × 3. Multiply the numbers of the first row of the first matrix with the corresponding numbers of the first column of the second matrix and add them up, this will be the value in the first row of the first column in the final answer. Multiply the numbers of the first row of the first matrix with the corresponding numbers of the second column of the second matrix and add them up, this will be the value in the first row of the second column in the final answer. Repeat for the second row of the first matrix. Final answer. Object 2×3 × 3×2 Matrices are conformable and final matrix has order of 2 × 2 6 × 4 + 4 × 7 + 8 × 6 = 100 6 × 2 + 4 × 4 + 8 × 3 = 52 10 × 4 + 5 × 7 + 1 × 6 = 81 10 × 2 + 5 × 4 + 1 × 3 = 43 100 52 [ ] 81 43 Image The table below summarises some common transformation vectors and what they mean. 0 1 [ ] 1 0 1 0 [ ] 0 −1 −1 0 [ ] 0 1 cos2π sin2π [ ] sin2π cos2π π 0 [ ] 0 1 1 0 [ ] 0 π π 0 [ ] 0 π Reflection in the line π¦ = π₯ Reflection in the π₯ axis Reflection in the π¦ axis Reflection in the line π¦ = (tanπ)π₯ Stretch with scale factor π parallel to π₯ axis Stretch with scale factor π parallel to π¦ axis Enlargement with scale factor π with origin as centre −1 0 [ ] 0 −1 0 1 [ ] −1 0 0 −1 [ ] 1 0 cosπ −sinπ [ ] sinπ cosπ 1 π [ ] 0 1 1 0 [ ] π 1 180° rotation about the origin 90° rotation clockwise about the origin 270° rotation clockwise about the origin π° rotation anticlockwise about the origin Shear with π₯ axis fixed Shear with π¦ axis fixed Matrices can also be multiplied by a scalar number. In that case, just multiply each element of the matrix by the scalar number. When an object undergoes two successive transformations π and π, a single transformation matrix, π × π can be found. Note that the matrices are multiplied in reverse order. 4 2 Example 3: Evaluate the following: 4 [7 4] 6 3 Example 4: A triangle with vertices at the coordinates π΄(1,3), π΅(1,7) and πΆ(2,6) is reflected in the π₯ axis and then rotated 45° anticlockwise about the centre of the origin. Find the coordinates of its image. Multiply each number in the matrix by 4. Final answer. 4×4 [4 × 7 4×6 16 [28 24 4×2 4 × 4] 4×3 8 16] 12 In 3 dimensions, transformation matrices are 3 × 3 and can be found in a similar way as in 2 dimensions, using 1 0 0 the unit vectors π = [0], π = [1] and π = [0]. 0 0 1 1 0 0 [0 1 0 ] 0 0 −1 Reflection in the π₯π¦ plane or π§ = 0 1 0 [0 cosπ 0 sinπ 1 0 0 [0 −1 0] 0 0 1 Reflection in the π₯π§ plane or π¦ = 0 [ −1 0 0 [ 0 1 0] 0 0 1 Reflection in the π¦π§ plane or π₯ = 0 cosπ [ sinπ 0 cosπ 0 −sinπ 0 −sinπ] cosπ Rotation of π° about π₯ axis 0 sinπ 1 0 ] 0 cosπ Rotation of π° about π¦ axis −sinπ cosπ 0 0 0] 1 Rotation of π° about π§ axis Invariant Points and Lines of a Linear Transformation 4 2 6 4 8 Example 2: Evaluate the following matrix multiplication: [ ] × [7 4] 10 5 1 6 3 Check the order of matrices. Multiply the combined transformation matrix with the position vector of the object. Write down the coordinates of each vector. 1 0 0 πΌ = [0 1 0] 0 0 1 Matrices as Transformations All matrices are 2 × 2 so they are conformable. 13 8 [ ] 14 14 11 8 [ ] 8 3 Check if all matrices have the same order. Add numbers in corresponding position e.g. 7 + 6 = 13. 0 0 [ ] 0 0 Identity matrices are often denoted by πΌ and are always square matrices. The diagonal elements from top left to bottom right are always ones, and all other elements are always zeros. When a matrix is multiplied with the identity matrix of the same order, the product should equal itself. Here are two examples of identity matrices: Matrices with the same order can be added together or subtracted from one another, by adding or subtracting elements in the corresponding positions. These are known as conformable matrices. Example 1: Evaluate the following matrix sum: [ 0 [0] 0 √2 −√2 √2 √2 2 2 × [1 0 ] = 2 2 0 −1 √2 √2 √2 −√2 [2 [2 2 ] 2 ] √2 √2 2 2 × [1 1 2] 3 7 6 √2 −√2 [2 2 ] 4√2 = [ 2√2 4√2 ] −√2 −3√2 −2√2 π΄(2√2, −√2), π΅(4√2, −3√2), πΆ(4√2, −√2) Find the transformation matrix for reflection in the π₯ axis. Find the transformation matrix for 45° anticlockwise rotation about the centre of origin. https://bit.ly/pmt-edu 1 0 [ ] 0 −1 √2 −√2 cos45 −sin45 2 [ ]= 2 sin45 cos45 √2 √2 [2 2 ] A point is invariant if its image after transformation is mapped onto itself. A line of invariant points is a line which all points map onto themselves. On the other hand, an invariant line consists of points which are mapped onto any point of the line - not necessarily the object point. 3 0 1 Example 5: Show that [ ] is an invariant point for the transformation [ ]. Find the line of invariant for 3 −1 2 this transformation. 0 1 3 Multiply [ ] × [ ] to find the image. 3 −1 2 Use the conditions of invariant points to form two simultaneous equations. Simplify the equations to find the line of invariant points. 0+3 3 ]=[ ] −3 + 6 3 The object and image have the same matrices, so it is an invariant point. π₯ π₯ 0 1 [ ] × [π¦] = [π¦] −1 2 0π₯ + π¦ = π₯ −π₯ + 2π¦ = π¦ π¦=π₯ [ 0 1 Example 6: Find the invariant line for the transformation [ ]. −1 2 π₯ π₯′ 3 4 Equate [ ] × [π¦] = [ ] π¦′ 9 −2 Substitute π¦ = ππ₯ + π into both equations. Equate π¦′ = ππ₯′ + π. Solve for values of π and π for LHS to become 0. To satisfy the equation, (4π + 2)π must also be 0. Invariant lines. π₯ ′ = 3π₯ + 4π¦ π¦ ′ = 9π₯ − 2π¦ ′ π₯ = 3π₯ + 4ππ₯ + 4π π¦ ′ = 9π₯ − 2ππ₯ − 2π 9π₯ − 2ππ₯ − 2π = 3ππ₯ + 4π2 π₯ + 4ππ 0 = 4π2 π₯ + 5ππ₯ − 9π₯ + 4ππ + 2π = (4π2 + 5π − 9)π₯ + (4π + 2)π (4π + 9)(π − 1) = 0 or 4π + 2 = 0 −9 π= ,π = 1 4 1 Note: π = − is not applicable because 4π2 + 2 5π − 9 ≠ 0 π=0 9 π¦=− π₯ π¦=π₯ 4 https://bit.ly/pmt-cc https://bit.ly/pmt-cc