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Final

ECON 211 Study Guide - Final Guide: Elementary Matrix, Diagonal Matrix, Triangular Matrix


Department
Economics
Course Code
ECON 211
Professor
Mahdiyeh Entezarkheir
Study Guide
Final

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ECON 211 Exam Review
Derivatives
Product Rule
(f(x)g(x))=f(x)g(x)+f(x)g(x)
Quotient Rule
f(x)
g(x)
=f(x)g(x)f(x)g(x)
g(x)2
Power Rule
(f(x)n)=nf(x)n1f(x)
Chain Rule
dY
dZ =df
dX
dX
dZ
Economic Applications
Annual Interest (deposit A, interest r, n times a year for t years)
R=A·(1 + r
n)nt
System of Linear Equations
Gaussian Elimination
- Eliminate variables by substitution & elimination
Gauss-Jordanian Elimination
a11x1+a12x2+a13x3=b1
a21x1+a2x2+a23x3=b2
a31x1+a32x2+a33x3=b3
a11 a12 a13 b1
a21 a22 a23 b2
a31 a32 a33 b3
- Solve by using elementary row operations
Systems with Many Solutions
(1) (# of equations) < (# of variables) or
(2) (# of equations) = (# of variables) but some are linearly dependent
Systems with No Solution
(1) (# of equations) > (# of variables) or
(2) 2 “equal” equations that equate to different parameters
Matrix Algebra
The rank of a matrix is min(m,n)
Matrix Multiplication
ab
cd
ef
3×2AB
CD
2×3
=
aA +bC aB +bD
cA +dC cB +dD
eA +fC eB +fD
3×2
A&B are matrices, suppose AB exists, elements in AB are given by
(AB)i,j =
n
r=1
Ai,rBr,j
Identity matrix is a multiplicative identity, therefore AI = A and IB=B
Transpose of a Matrix
AB
CD
T
=AC
BD
Column matrix (dimension k×1), row matrix (1×n), diagonal matrix has
non zero elements on its main diagonal and zero everywhere else, upper-
triangular matrix has non-zero elements on its main diagonal and above
and zeroes everywhere below main diagonal (opposite is lower-
triangular matrix), idempotent matrix remains the same after multiplied
by itself (A×A = A), symmetric matrix is such that AT = A, square
matrix is when An×k where n=k, nonsingular matrix is a square matrix
with rank = # of rows/columns, it represents a system with a unique
solution
POST MIDTERM 2 MATERIAL
Inverse of a Matrix
Inverse of matrix A is matrix A-1 such A-1A=AA-1=I, a matrix is only
invertible if it is nonsingular. To solve a matrix, we need to use the
inverse of a matrix - write the system in the matrix format or Ax = b
the solution would be x = A-1b
Gauss-Jordanian Method
A=
abc
def
ghi
Rewrite matrix A as (A:I) or
abc100
def010
ghi001
Using elementary operations, we try to obtain (I:A-1) or
100ABC
010DEF
001GH I
A1=
ABC
DEF
GH I
Elementary Row Operations
i. Interchange two rows of a matrix
ii. Multiply each element in a row by the same nonzero number
iii. Change a row by adding it a multiple of another row
Properties of the Inverse of a Matrix
Assume A and B are 2 matrices, if A and B are invertible, then
(A-1)-1 = A
(A-1)T = (AT)-1
(AB)-1 = B-1A-1
Inverse of an 2×2 Matrix
A1=d
adbc b
adbc
c
adbc
a
adbc
A1=1
det(A)db
ca
Sarrus’ Rule (Quick Method) - Determinant for 3×3 Matrix
Write determinant twice, omit last column of 2nd determinant
Multiply from top down diagonally, subtract the sum with sum of
multiplying top down reverse diagonals
(a11a22a33 +a12a23a31 +a13a21a32)(a13a22a31 +a11a23a32 +a12a21a33)
Determinant of Triangular & Diagonal Matrices
The determinant of lower-triangular, upper-triangular, or diagonal
matrices is simply the product of their diagonal entries
Properties of Determinants
det AT = det A
det (AB) = (det A)(det B)
det (A+B) (det A) + (det B)
Cofactor Matrix & Adjoint
For any n×n matrix A, we can replace element aij by the sub-matrix
obtained by deleting row i and column j from A, and multiply each
element by (-1)i+j
Now replace each element cij by cji, we obtain the adjoint of A or adjA
Inverse of a Matrix using Determinants is
A1=1
|A|(adjA)
We can multiply the inverse of the cofactor matrix with b to get the
solutions for a system of equations
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