Lecture 16, Unit 12: Matrix Algebra & Unit 10:Returns to scale, homogenous functions andpartial elasticities6/5/2016
Outline1Solving systems of linear equationsMatrix inversionCramer’s ruleExamples2Unit 10Returns to scaleHomogeneityPartial elasticities3Summary
Objectives - Unit 12•Use matrix inversion to solve systems of linear equations•Use Cramer’s rule to solve systems of linear equations
Literature - Unit 12Renshaw, ch. 19
Solving systems of linear equations•Suppose that we have a linear system of equations in the formAx=b, and suppose that we have to solve forx•Note:Ais a n by n matrix of coefficients,xis a column vectorcontaining the variables to be solved andbis a column vectorcontaining constants•Two methods•Matrix inversion•Cramer’s rule•NB! Before using either method to solve the system, make sure thatall variables that you have to solve for are on the LHS in everyequation, while everything else appears on the RHS!
Solving systems of equations: matrix inversion•With matrix inversion, we can solve for axinAx=bby finding theinverse of the matrixA, and multiplying this with b, i.e.x=A-1b,A-1=1Det(A)adj(A)
Solving systems of equations: Cramer’s rule•Suppose thatAx=b. Also suppose that we have to solve forx•Instead of matrix inversion, we can also use Cramer’s rule to solveforx•According to Cramer’s rule,xi=Det(Ai)Det(A)•Det(Ai) is the determinant of the matrixAi, whereAiis a matrixthat is formed by replacing the ith column ofAwithb•A1replace first column ofAwithb•A2replace second column ofAwithb•A3replace third column ofAwithb•So, if there are two unknowns in two equations, you need threedeterminants (A,A1,A2); if there are three equations in threeunknowns, you need four determinants (A,A1,A2,A3)Do the examples on p. 591 and 593
ExampleConsumption, investment and government expenditure areC= 80 + 0.6Y,I= 100-150r,G= 29. Furthermore, money supply and