Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 1) In this investigation, you learned how to multiply a row matrix by another matrix and how to interpret the product. [1 x 4] x [4 x 2] = [1 x 2] 3 7 [3 6 -5 9] 3 6 5 8 = [90 171] - 9 7 To multiply matrices the number of columns in the first matrix must be the same as the number of rows in the second matrix 3(3) + 6(7) + (-5)(3) + 9(6) = 90 3(5) + 6(8) + (-5)(-9) + 9(7) = 171 The result is the number of rows of the first matrix and the number of columns in the second matrix Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 2) Matrix multiplication can be useful but only in certain situations. [2 x 3] x [3 x 2] = [2 x 2] 1 2 1 2 3 22 28 4 5 6 3 4 49 64 5 6 1(1) + 2(3) + 3(5) = 22 1(2) + 2(4) + 3(6) = 28 4(1) + 5(3) + 6(5) = 22 4(2) + 5(4) + 6(6) = 28 Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 3) In this investigation, you explored how powers of an adjacency matrix for a digraph and sums of the powers could be used to analyze the digraph and the situation it models. Squaring a Matrix 2 Multiplying a matrix by itself 1 2 1 2 1 2 7 10 3 4 3 4 3 4 12 22 1(1) + 2(3) = 7 1(2) + 2(4) = 10 3(1) + 4(3) = 15 3(2) + 4(4) = 22 Math 2/Unit 2/Lesson 3/ TOOLKIT: Matrix Methods (Investigation 1) In this investigation, you examined properties of matrices and their operations and compared them with corresponding properties of real numbers. Adding/Subtracting Matrices When you add or subtract matrices, the sizes must be exactly the same. [2 x 2] + [2 x 2] = [2 x 2] 1 2 5 6 6 8 3 4 7 8 10 12 Adding Matrices is Commutative 1 2 5 6 5 6 1 2 3 4 7 8 7 8 3 4 Additive Identity or Zero Matrix is a matrix with all zeros 1 2 0 3 4 0 Additive Inverse Matrix is the matrix that adds to the original matrix to create a zero matrix 1 2 1 - 2 0 3 4 - 3 - 4 0 Multiplicative Identity is the 1 0 matrix. 0 1 This matrix has all ones on the main diagonal and zeros everywhere else. It also must be a square matrix and it is commutative. 0 1 2 0 3 4 1 2 1 0 1 2 3 4 0 1 3 4 1 0 1 2 1 2 0 1 3 4 3 4 0 0 Inverse Matrix of A, where d - b A= is - c a 1 d - b A 1 ad bc - c a 5 3 If A= , then 4 4 4 - 3 1 A 1 5(4) 3(4) - 4 5 1 4 - 3 A 1 20 16 - 4 5 1 4 - 3 4 - 4 5 -3 1 4 1 A - 1 5 4 A 1 To find A-1 with your calculator use the x-1 button If you multiply [A] [A-1] you should get the identity matrix 1 0 0 1 -3 1 4 5 3 1 If A= then and A 4 4 - 1 5 4 -3 1 5 3 4 = 1 0 4 4 - 1 5 0 1 4