Matrices
Matrices Introduction to matricesAddition & subtractionScalar multiplicationMatrix multiplicationThe unit matrixMatrix division - the inverse matrixUsing matrices - simultaneous equationsMatrix transformationsVCE Maths Methods - Unit 2 - Matrices1
Matrices A matrix is an array of individual elements. The order (dimensions) of a matrix is defined by the number of rows &columns. 1 2 2 x 2 matrix 4 6 3 0 3 1 x 3 matrix 2459 4 x 1 matrixrows columns orderVCE Maths Methods - Unit 2 - Matrices2
Examples of matrices The daily rate for hiring cars:1 day2 - 7 days8 daysKia Rio 120 105 90Toyota Camry 140 125 110Holden Statesman 170 145 120 120 105 90 R 140 125 110 170 145 120 VCE Maths Methods - Unit 2 - Matrices3
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 10 holiday surcharge applied to the car rental: 120 105 90 10 10 10 130 115 100 R 140 125 110 S 10 10 10 R S 150 135 120 170 145 120 10 10 10 180 155 130 VCE Maths Methods - Unit 2 - Matrices4
Scalar multiplication All elements can be multiplied by a scalar (single number). eg a 20% increase in the cost of hire cars: 120 105 90 R old 140 125 110 170 145 120 R new 1.2 R old 144 126 108 R new 168 150 132 214 174 144 VCE Maths Methods - Unit 2 - Matrices5
Matrix multiplication AxB C Rows in the first matrix multiply by the columns in the second. The number of rows in A & the number of columns in B gives the dimensionsof C . The number of columns in A must match the number of rows in B. (m x n) (n x p) gives an (m x p) matrix. In general, B x A C. 3 2 4 [(2 3) (4 5)] [26 ] 5 VCE Maths Methods - Unit 2 - Matrices6
Matrix multiplication Rows multiply by columns: The number of rows in A & the number ofcolumns in B gives the dimension of C . The number of columns in A must match the number of rows in B. aa1211 a21 a22 bb1211 b21 b22 (a b ) (a b ) (a b ) (a b )122111121222 11 11 (a21 b11 ) (a22 b21 ) (a21 b12 ) (a22 b22 ) 1 4 1 (3 4) (0 ) (3 1) (0 3) 3 0 2 1 3 1 1 2 (1 4) ( 2 ) (1 1) ( 2 3) 22 12 0 3 0 12 3 4 1 1 6 3 5 VCE Maths Methods - Unit 2 - Matrices7
Possible matrix multiplications Rows multiply by columns: The number of rows in A & the number ofcolumns in B gives the dimension of C . The number of columns in A must match the number of rows in B. 1 2 2 5 -219 43 3 4 0 7 -62x22x22x2 0 0 6 3212 4 123x11x3VCE Maths Methods - Unit 2 - Matrices00 42 84 3x3 0 3 2 1 2 [8 ] 4 1x33x1 1x1 1 10 9 8 52 2 34 7 6 5 3 2x33x12x18
The unit matrix The unit matrix (I) is a square matrix that can be multiplied by anothermatrix (A) to not alter that matrix. AI IA A if A is a square matrix. Non square matrices can be multiplied by a square identity matrix. 2 3 1 0 (2 1) (3 0) (2 0) (3 1) 2 3 6 2 0 1 (6 1) (2 0) (6 0) (2 1) 6 2 1 0 2 3 2 3 0 1 6 2 6 2 1 0 0 4 5 0 450 010 3 6 6 0 0 1 3 6 6 VCE Maths Methods - Unit 2 - Matrices9
Matrix division - the inverse matrix A square matrix has an inverse matrix A-1 , where A x A-1 I. Multiplying by A-1 is equivalent to division. For a 2 x 2 matrix: 1 a b 1 d c d cad bc b a 2 x 2 Matrix determinant (det A) ad - bc If det A 0, no solution exists. If both rows of the matrix are multiples of each other, then thedeterminant will be zero. (A singular matrix)VCE Maths Methods - Unit 2 - Matrices10
Matrix division - the inverse matrix For example, the matrices shown below: 1 2 3 3 5 5 3 1 10 9 3 2 1 5 19 3 5 19 3 19 3 2 3 19 2 19 VCE Maths Methods - Unit 2 - Matrices 1 2 3 6 9 9 3 1 18 18 6 2 Det A 0, no solution exists.11
Using matrices - simultaneous equations Matrices can be used to help solve simultaneous equations of two or morevariables. For example, finding the equation of a quadratic curve (y ax2 bx c)that passes through three points (-1,6) , (0, 3) & (2, 9).a 1 6 bc1 1 1 a 6 0 0 1b3 4 2 1 c 9 11 1 1 6 a 0 0 1 3 b 4 2 1 9 c 2 3 1 6 a 4 3 1 3 b 0 6 0 9 c VCE Maths Methods - Unit 2 - Matrices6 a( 1)2 b( 1) c23 a(0) b(0) c29 a(2) b(2) c 2 a 1 b 3 c y 2x 2 x 312
Matrix transformations - translations Matrix 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.yx’ x a(5,3)(1,2)y’ y b 1 4y intercept: x 0x 1 4 5 2 1 3 x a x ' y y'b VCE Maths Methods - Unit 2 - Matrices13
Matrix transformations - reflectionsReflection around the y x line: the x & y co-ordinates are swapped.yx’ yy’ x(1,2)(2,1)y intercept: x 0x 0 1 1 1 0 2 (1 0) (1 2) 2 (1 1) (0 2) 1 0 1 x x ' 1 0 y y ' VCE Maths Methods - Unit 2 - Matrices14
Matrix transformations - reflectionsReflection around the x axis: y value changes sign.yx’ xy’ -y(1,2)y intercept: x 0(1,-2)x 1 0 1 0 1 2 (1 1) (0 2) (0 1) (1 2) 1 2 1 0 x x ' 0 1 y y ' VCE Maths Methods - Unit 2 - Matrices15
Matrix transformations - reflectionsReflection around the y axis: x value changes sign.yx’ -xy’ y(-1,2) (1,2)y intercept: x 0x 1 0 1 0 1 2 ( 1 1) (0 2) (0 1) (1 2) 1 2 1 0 x x ' y y ' 0 1 VCE Maths Methods - Unit 2 - Matrices16
Matrix transformations - dilationsDilation from the y axis: x value is multiplied.y(1,2)y intercept: x 0x’ 5x 5y’ y(5,2)x 5 0 1 0 1 2 (5 1) (0 2) (0 1) (1 2) 5 2 k 0 x x ' 0 1 y y ' VCE Maths Methods - Unit 2 - Matrices17
Matrix transformations - dilationsDilation from the x axis: y value is multiplied.y (1,4)x’ xy’ 2y 4(1,2)y intercept: x 0x 1 0 1 0 2 2 (1 1) (0 2) (0 1) (2 2) 1 4 1 0 x x ' 0 k y y ' VCE Maths Methods - Unit 2 - Matrices18
Matrix transformations - rotationsAnti-clockwise rotation about the origin.yx’ cos90 x - sin90 yy’ sin90 x cos90 y(1,2)(-2,1)y intercept: x 0x cos90 sin90 1 sin90 cos90 2 (0 1) (1 2) (1 1) (0 2) cosθ sinθ VCE Maths Methods - Unit 2 - Matrices 2 1 sinθ x x ' y y ' cosθ 19
VCE Maths Methods - Unit 2 - Matrices Addition & subtraction of matrices 4 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 10 holiday surcharge applied to the car rental: R .
22 Dense matrices over the Real Double Field using NumPy435 23 Dense matrices over GF(2) using the M4RI library437 24 Dense matrices over F 2 for 2 16 using the M4RIE library447 25 Dense matrices over Z/ Z for 223 using LinBox’s Modular double 455 26 Dense matrices over Z/ Z for 211 using LinBox’s Modular&l
Class XII – NCERT – Maths Chapter 3 - Matrices 3.Matrices . Exercise 3.1 . Question 1: In the matrix 2 5 19 7 5 35 2 12 2 3 1 5 17. A . As the given matrices are equal, their corresponding elements are also equal. Class XII – NCERT – Maths . Chapter 3 - Matrices . 3.Matrices . Comparing the corresponding elements, we get: .
matrices with capital letters, like A, B, etc, although we will sometimes use lower case letters for one dimensional matrices (ie: 1 m or n 1 matrices). One dimensional matrices are often called vectors, as in row vector for a n 1 matrix or column vector for a 1 m matrix but we are going
Matrices and Determinants (i) Matrices Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a . Further Reduced ISC Class 12 Maths Syllabus 2020-21
of freedom involve spectral analysis of matrices. The discrete Fourier transform, including the fast Fourier transform, makes use of Toeplitz matrices. Statistics is widely based on correlation matrices. The generalized inverse is involved in least-squares approximation. Symmetric matrices are inertia, deformation, or viscous tensors in
BLOSUM vs. PAM Equivalent PAM and BLOSUM matrices based on relative entropy PAM100 Blosum90 PAM120 Blosum80 PAM160 Blosum60 PAM200 Blosum52 PAM250 Blosum45 PAM matrices have lower expected scores for the BLOSUM matrices with the same entropy BLOSUM matrices “generally perform better” than PAM matrices
SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS PART A: MATRICES A matrix is basically an organized box (or “array”) of numbers (or other expressions). In this chapter, we will typically assume that our matrices contain only numbers. Example Here is a matrix of size 2 3 (“2 by 3”), because it has 2 rows and 3 columns: 10 2 015
In this week’s lectures, we learn about matrices. Matrices are rectangular arrays of numbers or other mathematical objects and are fundamental to engineering mathematics. We will define matrices and how to add and multiply them, discuss some special matrices such as the identity and zero matrix,
