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

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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)n−1f�(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

I3x3=

100

010

001

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

A−1=

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

A−1=d

ad−bc −b

ad−bc

−c

ad−bc

a

ad−bc

A−1=1

det(A)d−b

−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

A−1=1

|A|(adjA)

We can multiply the inverse of the cofactor matrix with b to get the

solutions for a system of equations