Matrices Introduction to matrices Addition & subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations
Matrices A matri is an arra of individual elements. The order (dimensions) of a matri is defined b the number of rows & columns. 2 4 6 3 0 3 2 2 matri 2 4 5 9 3 matri 4 matri rows columns order 2
Eamples of matrices The dail rate for hiring cars: da 2-7 das 8 + das Kia Rio $20 $05 $90 Toota Camr $40 $25 $0 Holden Statesman $70 $45 $20 R 20 05 90 40 25 0 70 45 20 3
Addition & subtraction of matrices A + B C A & B must be of the same order. Corresponding elements in A & B are added or subtracted. C has the same order as A & B. The commutative law holds for matrices: A + B B + A eg a $0 holida surcharge applied to the car rental: R 20 05 90 40 25 0 70 45 20 S 0 0 0 0 0 0 0 0 0 R +S 30 5 00 50 35 20 80 55 30 4
Scalar multiplication All elements can be multiplied b a scalar (single number). eg a 20% increase in the cost of hire cars: R old 20 05 90 40 25 0 70 45 20 R new R new.2 R old 44 26 08 68 50 32 24 74 44 5
Matri multiplication A B C Rows in the first matri multipl b the columns in the second. The number of rows in A & the number of columns in B gives the dimensions of C. The number of columns in A must match the number of rows in B. (m n) (n p) gives an (m p) matri. In general, B A C. 2 4 3 5 (2 3)+ (4 5) [ ] 26 [ ] 6
Matri multiplication Rows multipl b columns: The number of rows in A & the number of columns in B gives the dimension of C. The number of columns in A must match the number of rows in B. a a 2 a 2 a 22 3 0 2 b b 2 b 2 b 22 4 2 3 (a b )+(a 2 b 2 ) (a b 2 )+(a 2 b 22 ) (a 2 b )+(a 22 b 2 ) (a 2 b 2 )+(a 22 b 22 ) (3 4)+(0 2 ) (3 )+(0 3) ( 4)+( 2 2 ) ( )+( 2 3) 2+0 3+0 4 +6 2 3 3 5 7
Possible matri multiplications Rows multipl b columns: The number of rows in A & the number of columns in B gives the dimension of C. The number of columns in A must match the number of rows in B. 2 2 5 3 4 0 7 2 2 2 2 2 2-2 9-6 43 0 3 2 2 8_ 4 [ ] 3 3 0 2 4 3 2 _ 0 0 0 6 4 2 2 8 4 0 9 8 7 6 5 2 3 52 34 3 3 3 3 2 3 3 2 8
The unit matri The unit matri (I) is a square matri that can be multiplied b another matri (A) to not alter that matri. AI IA A if A is a square matri. Non square matrices can be multiplied b a square identit matri. 2 3 6 2 0 0 (2 )+(3 0) (2 0)+(3 ) 2 3 (6 )+(2 0) (6 0)+(2 ) 6 2 0 0 2 3 6 2 2 3 6 2 4 5 0 3 6 6 0 0 0 0 0 0 4 5 0 3 6 6 9
Matri division - the inverse matri A square matri has an inverse matri A -, where A A - I. Multipling b A - is equivalent to division. For a 2 2 matri: a c b d ad bc d c b a 2 2 Matri determinant (det A) ad - bc If det A 0, no solution eists. If both rows of the matri are multiples of each other, then the determinant will be zero. (A singular matri) 0
Matri division - the inverse matri For eample, the matrices shown below: 9 2 3 3 5 0 9 5 9 3 9 5 3 3 2 5 3 3 2 3 9 2 9 8 8 2 3 6 9 Det A 0, no solution eists. 9 3 6 2
Using matrices - simultaneous equations 2 Matrices can be used to help solve simultaneous equations of two or more variables. For eample, finding the equation of a quadratic curve ( a 2 + b +c) that passes through three points (-,6), (0, 3) & (2, 9). 0 0 4 2 a b c 6 3 9 6 a( ) 2 +b( )+c 3 a(0) 2 +b(0)+c 9 a(2) 2 +b(2)+c 0 0 4 2 6 3 9 a b c 6 2 3 4 3 0 6 0 6 3 9 a b c 2 3 a b c 2 2 +3 a b c
Matri transformations - translations Matri operations can be used to find the transformations of points. These can be translations, reflections, rotations or dilations. Translations: The point can be moved across or up / down. intercept: 0 (,2) +4 (5,3) + 2 + a + b + 4 5 3 + a b ' ' 3
Matri transformations - reflections Reflection around the line: the & co-ordinates are swapped. (,2) intercept: 0 (2,) 0 0 2 ( 0)+( 2) ( )+(0 2) 2 0 0 ' ' 4
Matri transformations - reflections Reflection around the ais: value changes sign. (,2) - intercept: 0 0 0 2 (,-2) ( )+ (0 2) (0 ) ( 2) 2 0 0 ' ' 5
Matri transformations - reflections Reflection around the ais: value changes sign. (-,2) (,2) - intercept: 0 0 0 2 ( )+ (0 2) (0 )+ ( 2) 2 0 0 ' ' 6
Matri transformations - dilations Dilation from the ais: value is multiplied. (,2) (5,2) 5 5 intercept: 0 5 0 0 2 (5 )+ (0 2) (0 )+ ( 2) 5 2 k 0 0 ' ' 7
Matri transformations - dilations Dilation from the ais: value is multiplied. (,4) 2 4 intercept: 0 (,2) 0 0 2 2 ( )+ (0 2) (0 )+ (2 2) 4 0 0 k ' ' 8
Matri transformations - rotations Anti-clockwise rotation about the origin. (-2,) intercept: 0 (,2) cos90 - sin90 sin90 + cos90 cos90 sin90 sin90 cos90 2 (0 ) ( 2) ( )+ (0 2) 2 cosθ sinθ sinθ cosθ ' ' 9