lecture16Student(1).pdf - Lecture 16 Unit 12 Matrix Algebra Unit 10 Returns to scale homogenous functions and partial elasticities Outline 1 Solving

lecture16Student(1).pdf - Lecture 16 Unit 12 Matrix Algebra...

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Lecture 16, Unit 12: Matrix Algebra & Unit 10: Returns to scale, homogenous functions and partial elasticities 6/5/2016
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Outline 1 Solving systems of linear equations Matrix inversion Cramer’s rule Examples 2 Unit 10 Returns to scale Homogeneity Partial elasticities 3 Summary
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Objectives - Unit 12 Use matrix inversion to solve systems of linear equations Use Cramer’s rule to solve systems of linear equations
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Literature - Unit 12 Renshaw, ch. 19
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Solving systems of linear equations Suppose that we have a linear system of equations in the form Ax = b , and suppose that we have to solve for x Note: A is a n by n matrix of coefficients, x is a column vector containing the variables to be solved and b is a column vector containing constants Two methods Matrix inversion Cramer’s rule NB! Before using either method to solve the system, make sure that all variables that you have to solve for are on the LHS in every equation, while everything else appears on the RHS!
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Solving systems of equations: matrix inversion With matrix inversion, we can solve for a x in Ax = b by finding the inverse of the matrix A , and multiplying this with b, i.e. x = A - 1 b , A - 1 = 1 Det ( A ) adj ( A )
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Solving systems of equations: Cramer’s rule Suppose that Ax = b . Also suppose that we have to solve for x Instead of matrix inversion, we can also use Cramer’s rule to solve for x According to Cramer’s rule, x i = Det ( A i ) Det ( A ) Det ( A i ) is the determinant of the matrix A i , where A i is a matrix that is formed by replacing the ith column of A with b A 1 replace first column of A with b A 2 replace second column of A with b A 3 replace third column of A with b So, if there are two unknowns in two equations, you need three determinants ( A , A 1 , A 2 ); if there are three equations in three unknowns, you need four determinants ( A , A 1 , A 2 , A 3 ) Do the examples on p. 591 and 593
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Example Consumption, investment and government expenditure are C = 80 + 0 . 6 Y , I = 100 - 150 r , G = 29. Furthermore, money supply and
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