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International Information Technology University Department of Information System and Mathematical Modeling Mathematic in Economic Lesson 3 System of m linear equation with n variables Ph. D Alipova B. N. Mathematic in Economic / Ph. D Alipova B. N. 1

Input-Output Analysis of W. Leontief (1) Mathematic in Economic / Ph. D Alipova B. N. 2

Input-Output Analysis of W. Leontief (2) Mathematic in Economic / Ph. D Alipova B. N. 3

Example (Application to the Economic – Input-Output Analysis of W. Leontief) Mathematic in Economic / Ph. D Alipova B. N. 4

Trace of matrix A (m=n) is the sum of diagonal elements A= 1. tr A = tr A 2. If D is diagonal matrix with elements , so for ∀ m∊ℝ 3. tr (AB) = tr (BA) (but AB≠BA) 4. If C is non-singular matrix of dimension n, then tr (C ¹AC)= tr A Properties of tr A: 1. tr A=tr A’ 2. If D with (i=1, 2, …, n), f or every m tr (D‴)= 3. If A, B (m=n), so tr (AB)=tr (BA) 4. If , tr (Cˉ¹ AC)=tr A Mathematic in Economic / Ph. D Alipova B. N. 5

Rank of a matrix (1) = Mathematic in Economic / Ph. D Alipova B. N. 6

Rank of a matrix (2) Rank of matrix is not changed by: 1. 2. 3. 4. 5. Eliminating of zero row (column) Multiplying the row (column) by a non-zero constant Interchanging of rows (columns) Adding a multiple of one row (column) to the another row (column) Transpose Example Let we have we can denote But Solution: Mathematic in Economic / Ph. D Alipova B. N. 7

Theorem of Kronecker-Capelli: System of m linear equations of the form is consistent, only if r

Cramer’s Rule (8) (9) Mathematic in Economic / Ph. D Alipova B. N. 9

Example given earlier Mathematic in Economic / Ph. D Alipova B. N. (Inverse) 10

Gaussian Elimination Mathematic in Economic / Ph. D Alipova B. N. 11

Gaussian Elimination (Example) Mathematic in Economic / Ph. D Alipova B. N. 12

Jordan-Gauss Method Step 1. Pick solving element in augmented matrix (preferably “ 1”) Step 2. Division of all the rest elements in solving row by solving element. Step 3. Name all the rest elements of solving column as “ 0” Step 4. Calculate all the rest elements of augmented matrix (except solving row and solving column) as follows: (rule of rectangle) -solving element Step 5. Pick new solving element in augmented matrix (preferably “ 1”) except former row Step 6. Repeat Step 2 -4 Jordan-Gauss Method can be used as for solving of system of equation, as for finding of inverse Matrix (A I) (augmented matrix with identity matrix) Mathematic in Economic / Ph. D Alipova B. N. 13

Next Lesson is Elements of vector algebra Mathematic in Economic / Ph. D Alipova B. N. 14