[PDF] Topic 3: MATRICES PDF Topic3.pdf

Content • Adding, Subtracting and Multiplying Matrices • Matrix Inversion • Example: c12 = (2x2) + (3x3) + (4x4) = 29 Calculate the determinant of this as:

Previous PDF

Next PDF

[PDF] Matrix Multiplication

2 nov 2005 · tion and subtraction of matrices, as well as scalar multiplication, were introduced For each of the cases below, calculate AB (Click on the

[PDF] Basic Matrix Manipulation with a TI-89/TI-92/Voyage 200

The calculator will return an error if the dimensions are not the same ) Multiplying Matrices: Matrix multiplication is easy on the TI-89 For scalar multiplication,

[PDF] Calculator Instructions for Matrices - Math 4 Tigers

6 mar 2021 · Use the blue option above the multiplication symbol to choose a matrix • If the matrix is bigger than a 2x2, the calculator will ask you to enter

[PDF] Quiz Review Answer Keypdf

Matrix Multiplication Worksheet 2x2 3xz 14219 04/2) 2x2 Pl(o) + (-3)4 2 1579 (1)+(-3) (-2) (0) + 2(4) L8 #Jocs Solve for X 2x2 by hand, 3x3 with calculator

[PDF] Topic 3: MATRICES

Content • Adding, Subtracting and Multiplying Matrices • Matrix Inversion • Example: c12 = (2x2) + (3x3) + (4x4) = 29 Calculate the determinant of this as:

[PDF] fx-991EX/fx-570EX - CASIO

Operations with matrices and matrix related calculations are all located in the Matrix Menu From the Main Menu, multiplication, follow these simple commands To recall the without scrolling To calculate the determinant of the matrix, press

[PDF] The inverse of a 2x2 matrix - Mathcentre

Once you know how to multiply matrices it is natural to ask whether they can be divided The Suppose we calculate the product of the two matrices ( 4 3

[PDF] 41 Matrix Multiplication

columns The Matrix A is a 2x2 matrix, b is a 1x3 matrix 4 1 Matrix multiplication: Details result, the Frisch-Waugh (1933) Theorem: To calculate b2 (or b1)

[PDF] 2-2 Properties of Matrices ANSWERS - Mayfield City Schools

I can multiply two matrices using the graphing calculator I can use matrix 2x2 2x3 → What do you notice about the resulting product matrices? What you end

Topic 3: MATRICES

Jacques (3rd Edition):

Chapter 7.1- 7.2

Content

•Adding, Subtracting and Multiplying

Matrices

•Matrix Inversion •Example: Model of National Income

A Vector: list of numbers arran

g ed in a row or column e. g . consumption of 10 units X and 6 units of Y g ives a consumption vecto r (X,Y) of (10,6) (6,10)

A Matrix: a two-dimensional arra

y o f numbers arran g ed in rows an d columns e.g. A =

232221131211

aaaaaa a 2 X 3 matrix with 2 rows and 3 columns component a ij in the matrix is in the i th row and the j th column e.g. let a ij be amount g ood j consumed by individual i - columns1-3:represent goods X, Y& Z - rows 1-2:represent individuals 1 & 2

Matrix of consumption

C =

232221131211

cccccc

6045100

Individual 1 consumes 0 of X, 10 of Y

and 5 of Z

Individual 2 consumes 4 of X, 0 of Y an

6 of Z

NOTE

Row Vector is a matrix with only 1 row :

A = [5 4 3] 1 X 3 matrix

Column Vectoris a matrix with only 1

column : A = 3 45

3 X 1 matrix

Transposing Matrices

A =

232221131211

aaaaaa

2 X 3 matrix

Then A T

231322122111

aaaaaa

3X2 matrix

the transpose of a matrix replaces rowsb y columns. A=

6045100

then A T

6501040

Adding and Subtracting Matrices

Matrices must have same number o

f rows and columns, m X n

Just add (subtract) the corresponding

elements.....

A + B + C = D i.e. a

ij + b ij + c ij = d ij 56215
2331
6125
1439

A - B = E i.e. a

ij - b ij = e ij 5554
6125
1439

Multiplying Matrices To multiply A and B,

N o. Columns in A = No. Rows in B

Then A x B = C

(1x 3) (3x 2) = (1x 2) 1211

323122211211

131211

cc bbbbbb .aaa c 11 = (a 11 .b 11 )+ (a 12 .b 21
) +(a 13 .b 31
c 12 = (a 11 .b 12 )+ (a 12 .b 22
) +(a 13 .b 32
2925

423521

432
c 11 = (2x1) + (3x5) + (4x2) = 25 c 12 = (2x2) + (3x3) + (4x4) = 29

2423222114131211

114521011213

401012

cccccccc.

5617234527

c 11 = (2x3) + (1x1) + (0x5) = 7 c 12 = (2x1) + (1x0) + (0x4) = 2 c 13 = (2x2) + (1x1) + (0x1) = 5 c 14 = (2x1) + (1x2) + (0x1) = 4 c 21
= (1x3) + (0x1) + (4x5) = 23 c 22
= (1x1) + (0x0) + (4x4) = 17 c 23
= (1x2) + (0x1) + (4x1) = 6 c 24
= (1x1) + (0x2) + (4x1) = 5

SCALAR MULTIPLICATION

If A =

22211211

aaaa

Then 3A =

22211211

3333
aaaa A = 1234
then 2A = 2468

And 3A =

36912

Practice Transposing, Adding, Subtracting and

Multiplying Matrices using examples from any Text

Book - or simply by writing down some simple

matrices yourself....

Determinant of a Matrix If A =

dcba aaaa

22211211

Now we can find the determinant......

Multiply elements in any one row or any

one column by corresponding co-factors, and sum.....

Select row 1.... |A| = a

11 .C 11 + a 12 .C 12 ad - bc

Select column 2

|A| = a 12 .C 12 + a 22
.C 22
= b(-c)+da

MATRIX INVERSION

Square matrix: no. rows = no. columns

Identity Matrix I: AI = A and IA = A

I = 1001
(for 2 X 2 matrix)

Inverse Matrix A

-1 : A.A -1 = I A -1 .A= I T

O INVERT

2 X 2 M ATRIX

If A =

dcba

1) Get Cofactor Matrix:

abcd

2) Transpose Cofactor Matrix:

acbd 1) multiply matrix by |A|1 so acbd bcad1 (i.e. divide each element by ad- bc) If |A| =0 then there is no inverse......(matrix is singular)

Example....find the inverse of matrix A

A = 4321
|A| = ad-bc = (1.4)-(2.3) = -2 (non-singular) A -1 1324
21
21
23
12

Check : A.A

-1 = I = 1001

Example....find the inverse of matrix B

B = 10542
|B| = ad - bc = (2.10) - (4.5) = 0 therefore, matrix is singular and inverse does not exist

Example Expenditure model of national

income

Y = Income

C = Consumption

I = Investment

G = Government expenditure

Y = C+I+G (1)

The consumption function is

C = a + bY (2)

N ote C and Y are endogenous. I and G are exogenous.

How to solve for values of endogenous

variables Y and C?

Method 1

Solve the above equations directly,

substituting expression for C in eq. (2) into eq. (1)

Thus, Y = a + bY+I+G

Solve for Y as:

Y - bY = a + I + G

Y(1 - b) = a + I + G

Thus, bGIaY 1

Substitute this value for Y into eq. (2) an

d solve for C: babGI bGIabaC 1)( 1

Method 2 N

ow solve the same problem using matrix algebra:

Rewrite (1) and (2) with endogenous

variables, C and Y, on left hand side

From eq. 1: Y - C = I + G

From eq. 2: -bY + C = a

Now write this in matrix notation:

aGI CY b111 or A.X = B

We can solve for the endogenous variables

X, by calculating the inverse of the A

matrix and multiplying by B:

Since AX=B

X=A -1 B

To invert the 2 X 2 A matrix, recall the

steps from earlier in the lecture

If A =

dcba , then A -1 acbd bcad1

In this case, where

111
bA the determinant of A is : |A| = 1.1 - [- 1.- b] = 1 - b

Cofactor Matrix:

111b

Transpose Cofactor Matrix:

111
b

The inverse is :

bbbbb b bAquotesdbs_dbs14.pdfusesText_20

[PDF] [PDF] Topic 3: MATRICES

Topic 3: MATRICES

Jacques (3rd Edition):

Chapter 7.1- 7.2

Content

Matrices

A Vector: list of numbers arran

A Matrix: a two-dimensional arra

232221131211

Matrix of consumption

232221131211

6045100

Individual 1 consumes 0 of X, 10 of Y

Individual 2 consumes 4 of X, 0 of Y an

6 of Z

Row Vector is a matrix with only 1 row :

A = [5 4 3] 1 X 3 matrix

Column Vectoris a matrix with only 1

3 X 1 matrix

Transposing Matrices

232221131211

2 X 3 matrix

231322122111

3X2 matrix

6045100

6501040

Adding and Subtracting Matrices

Matrices must have same number o

Just add (subtract) the corresponding

A + B + C = D i.e. a

A - B = E i.e. a

Multiplying Matrices To multiply A and B,

Then A x B = C

323122211211

131211

423521

2423222114131211

114521011213

401012

5617234527

SCALAR MULTIPLICATION

If A =

22211211

Then 3A =

22211211

And 3A =

Practice Transposing, Adding, Subtracting and

Multiplying Matrices using examples from any Text

Book - or simply by writing down some simple

Determinant of a Matrix If A =

22211211

Now we can find the determinant......

Multiply elements in any one row or any

Select row 1.... |A| = a

Select column 2

MATRIX INVERSION

Square matrix: no. rows = no. columns

Identity Matrix I: AI = A and IA = A

Inverse Matrix A

O INVERT

If A =

1) Get Cofactor Matrix:

2) Transpose Cofactor Matrix:

Example....find the inverse of matrix A

Check : A.A

Example....find the inverse of matrix B

Example Expenditure model of national

Y = Income

C = Consumption

I = Investment

G = Government expenditure

Y = C+I+G (1)

The consumption function is

C = a + bY (2)

How to solve for values of endogenous

Method 1

Solve the above equations directly,

Thus, Y = a + bY+I+G

Solve for Y as:

Y - bY = a + I + G