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MiE-1-Matrix-placed.pptx

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International Information Technology University Department of Information Systems and Mathematical Modeling Mathematic in Economic International Information Technology University Department of Information Systems and Mathematical Modeling Mathematic in Economic Lesson 1 Matrix. Determinants. Ph. D Alipova B. N. Mathematic in Economic / Ph. D Alipova B. N. 1

Linear Equations The convention mentioned above is simply the rule concerning the multiplication of Linear Equations The convention mentioned above is simply the rule concerning the multiplication of matrices. These arrays of real numbers of the form Mathematic in Economic / Ph. D Alipova B. N. 2

Matrix Definition Mathematic in Economic / Ph. D Alipova B. N. 3 Matrix Definition Mathematic in Economic / Ph. D Alipova B. N. 3

Elementary Row Operations A system of m linear equations of the form (1) The Elementary Row Operations A system of m linear equations of the form (1) The system can be represented by array (2) Mathematic in Economic / Ph. D Alipova B. N. 4

Addition and subtraction F F The matrix F obtained from A by multiplying each Addition and subtraction F F The matrix F obtained from A by multiplying each entry of A by -1 Mathematic in Economic / Ph. D Alipova B. N. 5

Arithmetic rules with Matrices Mathematic in Economic / Ph. D Alipova B. N. 6 Arithmetic rules with Matrices Mathematic in Economic / Ph. D Alipova B. N. 6

Multiplication by a real number (Multiplication by scalar) Suppose that Mathematic in Economic / Multiplication by a real number (Multiplication by scalar) Suppose that Mathematic in Economic / Ph. D Alipova B. N. 7

Multiplication of Matrices Suppose that Mathematic in Economic / Ph. D Alipova B. N. Multiplication of Matrices Suppose that Mathematic in Economic / Ph. D Alipova B. N. 8

Transpose of a matrix Properties of transpose 1. (A’)’=A 2. (λA)’=λA’ 3. (A+B)’=A’+B’ 4. Transpose of a matrix Properties of transpose 1. (A’)’=A 2. (λA)’=λA’ 3. (A+B)’=A’+B’ 4. (AB)’=B’A’ Mathematic in Economic / Ph. D Alipova B. N. 9

Row Echelon Form Reduced Row Echelon Form 0 0 0 0 Mathematic in Economic Row Echelon Form Reduced Row Echelon Form 0 0 0 0 Mathematic in Economic / Ph. D Alipova B. N. 10

Illustration Mathematic in Economic / Ph. D Alipova B. N. 11 Illustration Mathematic in Economic / Ph. D Alipova B. N. 11

Determinant Matrix 2 x 2 Matrix 3 x 3 Rule of triangles or Rule Determinant Matrix 2 x 2 Matrix 3 x 3 Rule of triangles or Rule of Sarrus Matrix nxn Mathematic in Economic / Ph. D Alipova B. N. 13

Example Mathematic in Economic / Ph. D Alipova B. N. 14 Example Mathematic in Economic / Ph. D Alipova B. N. 14

Application to Linear Equation We now return to the problem mentioned above. Consider the Application to Linear Equation We now return to the problem mentioned above. Consider the two equations (consistent system) Mathematic in Economic / Ph. D Alipova B. N. 15

Example Mathematic in Economic / Ph. D Alipova B. N. 16 Example Mathematic in Economic / Ph. D Alipova B. N. 16

Row operations on determinants Column operations on determinants Mathematic in Economic / Ph. D Row operations on determinants Column operations on determinants Mathematic in Economic / Ph. D Alipova B. N. 17

Next Lesson is Inverse Matrix Mathematic in Economic / Ph. D Alipova B. N. Next Lesson is Inverse Matrix Mathematic in Economic / Ph. D Alipova B. N. 18