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Academic Skills Advice Matrices Summary To transpose a matrix write the rows as columns. For example: 2 1 5 A = [1 5 0 2 1 4 0 9] 2 2 1 5 0 AT = 1 5 0 9 2 1 4 2 To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers. For example: 2 5 −1 [4 1 2] + −3 3 4 3 [7 10 −2 3 5 4 2 ] = [11 5 −5 7 3 2 5 4] 8 −1 To multiply matrices the number of columns in the 1st must equal the number of rows in the 2nd because you multiply across the row and down the column. For example: (6 + 10) (−9 + 14) 3 2 2 −3 7 [4 6 ] × [ ] = [(8 + 30) (−12 + 42) 5 7 8 (2 − 25) (−3 − 35) 1 −5 16 = [ 38 −23 (21 + 16) (28 + 48)] (7 − 40) 5 37 30 76 ] −38 −33 Transformations When setting up the multiplication for a transformation the transformation matrix always goes first. Rotations: Clockwise: [ 𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜃 ] 𝑐𝑜𝑠𝜃 Reflections: 0 [ 1 1 ] : swaps 𝑥 𝑎𝑛𝑑 𝑦. 0 𝑐𝑜𝑠𝜃 Anticlockwise: [ 𝑠𝑖𝑛𝜃 −𝑠𝑖𝑛𝜃 ] 𝑐𝑜𝑠𝜃 Scaling / magnification: [ 3 0 ] : makes 𝑥 3x bigger and 𝑦 2x bigger 0 2 Composite: If a matrix A is reflected by T1 then magnified by T2 to give matrix B then the single transformation that would transform A into B is T 2T1 (in that order). H Jackson 2010 / Academic Skills 1 Except where otherwise noted, this work is licensed under http://creativecommons.org/licenses/by-nc-sa/3.0/ The Inverse (A-1): AA = I -1 1 (the identity matrix) e.g. [0 0 0 0 1 0] 0 1 or [ 1 0 ] 0 1 Finding the Inverse: 1 𝑎 If: A = [ 𝑐 A-1 = |𝐴| x adj 𝑏 ] 𝑑 |𝐴| = (𝑎𝑑) − (𝑏𝑐) Where: |𝐴| = determinant adj = adjoint adj = [ 𝑑 −𝑐 −𝑏 ] 𝑎 n.b. If det=0 matrix is singular and there is no inverse. If det≠0, matrix is non-singular. Solving Simultaneous Equations : If : then: 𝐴𝑥 = 𝐵 𝑥 = 𝐴−1 𝐵 where : 𝐴 is the matrix 𝑥 is the 𝑥, 𝑦, 𝑧 (𝑒𝑡𝑐) values 𝐵 is the answers All you need to remember to find the values of 𝑥, 𝑦, 𝑧 etc is to do: inverse x answers Example: We have: 5𝑥 − 6𝑦 = −8 −3𝑥 + 4𝑦 = 6 Solve and [ −8 5 −6 𝑥 ] [𝑦] = [ ] 6 −3 4 𝐴 So: simultaneously 𝑥 = 𝐵 𝑥 −8 [𝑦] = 𝐴−1 [ ] 6 𝑥 [𝑦 ] = H Jackson 2010 / Academic Skills 4 [ 2 3 1 1 4 6 −8 ] [ ]= [ ] 2 6 5 6 (Use the method shown above to find the inverse then multiply by the answers.) = 2 [ ] 3 2