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LINEAR MODELS AND MATRIX ALGEBRA - Part 3

LINEAR MODELS AND MATRIX ALGEBRA - Part 3. Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd edition. Identity Matrix. Characteristics: It is like the number 1: a(1) = (1)a=a. IA = AI = A. Identity Matrix.

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LINEAR MODELS AND MATRIX ALGEBRA - Part 3

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  1. LINEAR MODELS AND MATRIX ALGEBRA- Part 3 Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3rd edition

  2. Identity Matrix Characteristics: It is like the number 1: a(1) = (1)a=a IA = AI = A

  3. Identity Matrix Special Nature of I Matrices : it is possible to insert or delete an I w/o affecting the matrix product.

  4. Inverses and their Properties: • For a given matrix A, • The transpose is always derivable. • Its inverse matrix (another type of derived matrix) may or may not exist. • The inverse of matrix A is denoted by A-1 is defined only if A is a square matrix.

  5. Inverses and their Properties: Whether A is pre-multiplied or postmultiplied by A-1, the product will be the same identity matrix. This is an exception to the rule that matrix multiplication is not commutative

  6. Points to note • Not every square matrix has an inverse. Squareness is a necessary but not a sufficient condition for the existence of a matrix. • Non-singular matrix : if an inverse exists • Singular : if no inverse exists • If A-1 exists, then A is the inverse of A-1 just as A-1 is the inverse of A. A and A-1 are inverses of each other. • If A is nxn, then A-1 must also be nxn. • If an inverse exists, it is unique.

  7. AB=BA=I • Define C such that AC=CA=I • Premultiplying both sides of AB=I by C, • CAB=CI=C • Since CA=I, IB=C or B=C • That is, B and C must be one and the same inverse matrix.

  8. Solution of Linear Equation System The application of the concept of inverse matrix to the solution of simultaneous equation system is immediate and direct.

  9. Solution of Linear Equation System • One way of finding the solution of a linear equation system Ax=d, where the coefficient matrix is nonsingular, first find A-1, and then post multiply A-1 by the constant vector d. The product A-1d will then give the solution values of the variables.

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