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University of British Columbia
ECON 211
Mahdiyeh Entezarkheir

POST MIDTERM 2 MATERIAL ECON 211 Exam Review Inverse of a Matrix Derivatives Inverse of matAiis matriA-1suchA A=AA =I-1 , a matrix is only invertible if it is nonsingular. To solve a matrix, we need to use the Product Rule inverse of a matrix - write the system in the matriAx = bat or (f(x)g(x)) = f (x)g(x)+ f(x)g (x) -1 the solution would x =A b Quotient Rule Gaus-Jordanin Method f(x) = f (x)g(x) − f(x)g (x) ab c g(x) g(x)2 A =  de f  Power Rule gh i (f(x) ) = nf(x)−1f (x) ewrite matrix A A:I)or ab c 100 Chain Rule  de f 010  dY = df dX dZ dX dZ gh i 001 Economic Applications sing elementary oprations, we try to (I:A )or 100 AB C Annual Interest (depoA, interer,n times a year toyears)  010 DE F  R = A · (1 + )t n 001 GH I   System of Linear Equations AB C A−1=  DE F  Gaussian Elimination GH I - Eliminate variables by substitution & elimination Gauss-Jordanian Elimination  Elementary Row Operations a11 1 a 12 2a x13 3 1 a11 a12 a13 b1 i. Interchange two rows of a matrix a21 1 a 2 2 a 23 3b 2 ⇒ a21 a22 a23 b2 a x + a x + a x = b a a a b ii. Multiply each element in a row by the same nonzero number 31 1 32 2 33 3 3 31 32 33 3 iii. Change a row by adding it a multiple of another row - Solve by using elementary row operations Properties of the Inverse of a Matrix Systems with Many Solutions Assume A andB are 2 matricesA and B are invertible, then (1) (# of equations) < (# of variables) or (A ) =A1 (2) (# of equations) = (# of variables) but some are linearly d(A ) = (A ) -1 -1 -1-1 Systems with No Solution (AB) = B A (1) (# of equations) > (# of variables) or Invers ofdan 2×−bMtrix (2) 2 “equal” equations that equate to different parameters A −1= ad−bc ad−bc ad−bc ad−bc Matrix Algebra The rank of a matrix ism,n)( A −1= 1 d −b det(A) −ca Matrix Multiplication  ab  aA + bC aB + bD  Sarrus’ Rule (Quick Method) - Determinant for 3×3 Matrix AB  cd  CD = cA + dC cB + dD  Write determinant twice, omit last column of 2nd determinant ef 2×3 eA + fC eB + fD Multiply from top down diagonally, subtract the sum with sum of 3×2 3×2 multiplying top down reverse diagonals (a11 22 33 a12 23 31 a13 21 32 (a13 22 31 a11 23 32 a12 21 33 A &B are matrices, suppoAB exists, elementsABnare given by n Determinant of Triangular & Diagonal Matrices (AB) i,j Ai,r r,j The determinant of lower-triangular, upper-triangular, or diagonal r=1 matrices is simply the product of their diagonal entries Identiy matri is a multiplicative identiAI,=tAeandIBe=B 100 Properties of Determinants   detAT = detA I3x3= 010 001 det (AB) = (detA)(det B) det (A+B) ≠ (detA) + (det B) Transpose of a Matrix T AB = AC Cofactor Matrix & Adjoint CD BD For any n×n matrAx, we can replace elemanby the sub-matrix Column matrix (dimension k×1), row matrix (1×n), diagonal matrix hasned by deleting riand columnjfromA , and multiply each element by(-1)+j non zero elements on its main diagonal and zero everywhere else, upper- ij ji triangular matrix has non-zero elements on its main diagonal anNow replace each elemenc byc , we obtain the adjoAnoroadjA and zeroes everywhere below main diagonal (opposite is lower- triangular matrix), idempotent matrix remains the same after multipliedof a Matrix using Determinants is by itselA×A = A), symmetric matrix is such Ah=tA , square A−1= 1 (adjA) |A| matrix is wheA n×khere n=k, nonsingular matrix is a square matrix with rank = # of rows/columns, it represents a system with a uniquean multiply the inverse of the cofactor matbto get the solution solutions for a system of equations Crammer Rule Quadratic Form to Matrices The unique solution of a system Ax=B is Coefficient of X 2 is theithelement on the diagonal detB i i x i detA Coefficient of XXi j divided by 2 is the element (i, ) & j, ) i th Coefficient of XXi j is zero if the corresponding element does not exist where B is the matrixA with b replacing the i column of A , this is the unique solution if |A| does not equal to zero Optimization a11 1 a x12 2 x =13 3 1 Unconstrained Optimization a21 1 a x22 2 x =23 3 2 Quadratic function always equals to zero when x = 0 a31 1 a x32 2 x =33 3 3
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