Matrices Summary. To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers.

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1 Matrices Summary To transpose a matrix write the rows as columns. Academic Skills Advice For example: 2 1 A = [ ] A T = To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers. For example: [ ] + [ ] = [ 11 4 ] To multiply matrices the number of columns in the 1 st must equal the number of rows in the 2 nd because you multiply across the row and down the column. Transformations When setting up the multiplication for a transformation the transformation matrix always goes first. Rotations: Reflections: Composite: For example: 3 2 [ 4 6 ] [ (6 + 10) ( ) ( ) ] = [ (8 + 30) ( ) ( ) ] (2 2) ( 3 3) (7 40) = [ ] Clockwise: [ cosθ sinθ sinθ ] Anticlockwise: [cosθ sinθ cosθ sinθ cosθ ] Scaling / magnification: [ ] : swaps x and y. [3 ] : makes x 3x bigger and y 2x bigger If a matrix A is reflected by T 1 then magnified by T 2 to give matrix B then the single transformation that would transform A into B is T 2 T 1 (in that order). H Jackson 2010 /2013/ Academic Skills 1

2 The Inverse (A -1 ): AA -1 = I (the identity matrix) e.g. [ ] or [ ] Finding the Inverse: A -1 = 1 A x adj If: A = [a b c d ] Where: A = determinant adj = adjoint A = (ad) (bc) adj = [ d b c 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 : Ax = B where : then: x = A 1 B A is the matrix x is the x, y, z (etc) values B is the All you need to remember to find the values of x, y, z etc is to do: inverse x Example: Solve x 6y = 8 and 3x + 4y = 6 simultaneously We have: [ ] [x y ] = [ 8 6 ] A x = B So: [ x y ] = A 1 [ 8 6 ] (Use the method shown above to find the inverse then multiply by the.) [ x y ] = 1 2 [4 6 3 ] [ 8 6 ] = 1 2 [4 6 ] = [2 3 ] H Jackson 2010 /2013/ Academic Skills 2

3 3 X 3 Matrices Most operations are done exactly the same for a 3x3 matrix as for a 2x2. The formula for finding the inverse is the same but finding the determinant and the adjoint are a bit trickier. The Minor: The minor of each element is the determinant of everything that is not in the same row or column as the element. It is used when finding the determinant and the adjoint. e.g. Find the minor of each element in the top row of the matrix: [ d e f] To find the minor of element a cover up everything in the same row or column and the minor is the determinant of what is left: d e f the minor of a is: e f = ei fh h i To find the minor of element b cover up everything in the same row or column and the minor is the determinant of what is left: d e f the minor of b is: d f = di fg g i To find the minor of element c cover up everything in the same row or column and the minor is the determinant of what is left: d e f the minor of c is: d e = dh eg g h The signs: When finding the determinant or the adjoint we use alternate signs as follows: + + [ + ] + + H Jackson 2010 /2013/ Academic Skills 3

4 Finding the Inverse Remember: A -1 = 1 A x adj Where: A = determinant adj = adjoint Determinant: For the top row only, multiply each number by its minor and apply the alternate signs: The minor of each element in the top row If: A = [ d e f e f f e ] A = +a b d + c d h i g i g h Adjoint: The adjoint is the transpose of the cofactors (c T ) To find the cofactors you find the minor of each element (and use the alternate signs as shown above). If: A = [ d e f] The minors of each element e f f e Matrix of cofactors = + d + d h i g i g h b c c b + a h i g i a g h b c c b + a e f d f + a d e Nb. To find the adjoint we then need to transpose this matrix. H Jackson 2010 /2013/ Academic Skills 4

5 2 1 e.g. Find the inverse of the following matrix: A = [ 3 3 2] Determinant: A = = 2(1) 1( ) + ( 9) = 38 Cofactors = = [ ] [ ] Adjoint (C T ): = [ ] We have transposed the matrix of cofactors to get the adjoint. Inverse: A -1 = 1 A x adj = [ ] H Jackson 2010 /2013/ Academic Skills

6 Solving Simultaneous Equations Using Cramer s Rule: Cramers method is a way to solve simultaneous equations using matrices. e.g. Solve [ a c To find x: b d ] [x y ] = [e f ] to find x and y Replace 1 st column with answer column x = e b f d a b c d Determinant of matrix with 1st column replaced with the answer column. Determinant of original matrix To find y: y = a e c f a b c d Determinant of matrix with 2nd column replaced with the answer column. Determinant of original matrix Notice that the bottom of the fraction is always the determinant of the original matrix. e.g. Solve the simultaneous equations: 3x 4y = 10 x + 7y = Set up a matrix: [ ] [x y ] = [10 ] Determinant of matrix: = 2 Think of these as the. For help with finding the determinant, see Determinant Summary sheet. To find x: = 0 2 = 2 To find y: = = -1 H Jackson 2010 /2013/ Academic Skills 6

7 e.g. Solve the simultaneous equations: 2x + y 3z = 7 x + 3y + z = 6 x 4y z = 2 Think of these as the x 7 Set up the matrix: [ ] [ y] = [ 6 ] 1 4 z Determinant of matrix: = 1 4 To find x = = 1 To find y = 10 = 2 To find z = = -1 H Jackson 2010 /2013/ Academic Skills 7

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