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Final

# Exam Review Sheet All the equations & theories for the course.

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University of British Columbia

Economics

ECON 211

Mahdiyeh Entezarkheir

Fall

Description

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-Jordanin 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 oprations, 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
Identiy 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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