Matrix Algebra Basics

Pam Perlich

Urban Planning 5/6020

Algebra

Matrix

A

a11 ,, a1n

a21 ,, a2n

am1 ,, amn

Aij

A matrix is any doubly subscripted array of elements arranged in rows and columns.

Row Vector

[1 x n] matrix

 

jn aaaaA ,, 2 1

Column Vector

i

m

a

a

a

a

A 2

1

[m x 1] matrix

Square Matrix

B

5 4 7

3 6 1

2 1 3

Same number of rows and columns

The Identity

Identity Matrix

I

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Square matrix with ones on the diagonal and zeros elsewhere.

Transpose Matrix

A'

a11 a21 ,, am1

a12 a22 ,, am 2

a1n a2n ,, amn

Rows become columns and columns become rows

Matrix Addition and Subtraction

A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: 

Cij Aij Bij

Note: all three matrices are of the same dimension

Addition

A a11 a12

a21 a22

B b11 b12

b21 b22

C a11 b11 a12 b12

a21 b21 a 22 b22

If

and

then

Matrix Addition Example

A B 3 4

5 6

1 2

3 4

4 6

8 10

C

Matrix Subtraction

C = A - BIs defined by

Cij Aij Bij

Matrix Multiplication

Matrices A and B have these dimensions:

[r x c] and [s x d]

Matrix Multiplication

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s

Matrix Multiplication

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d

Computation: A x B = C

A a11 a12

a21 a22

B b11 b12 b13

b21 b22 b23

232213212222122121221121

2312131122121211 21121111

babababababa

babababababaC

[2 x 2]

[2 x 3]

[2 x 3]

Computation: A x B = C

A

2 3

1 1

1 0

and B

1 1 1

1 0 2

[3 x 2] [2 x 3]A and B can be multiplied

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

[3 x 3]

Computation: A x B = C

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

A

2 3

1 1

1 0

and B

1 1 1

1 0 2

[3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3

Inversion

Matrix Inversion

B 1B BB 1 I

Like a reciprocal in scalar math

Like the number one in scalar math

Linear System of Simultaneous Equations

1st Precinct : x1 x2 6

2nd Pr ecinct : 2x1 x2 9

First precinct: 6 arrests last week equally divided between felonies and misdemeanors.

Second precinct: 9 arrests - there were twice as many felonies as the first precinct.

Solution

9

6 *

1 2

1 1

2

1

x

x

3

3

2

1

x

x

1 2

1 1 Note: Inverse of is

1 2

1 1

9

6*

1 2

1 1 *

1 2

1 1*

1 2

1 1

2

1

x

x Premultiply both sides by inverse matrix

3

3 *

1 0

0 1

2

1

x

x A square matrix multiplied by its inverse results in the identity matrix.

A 2x2 identity matrix multiplied by the 2x1 matrix results in the original 2x1 matrix.

aijxj bi or Ax bj1

n

x A 1Ax A 1b

n equations in n variables:

unknown values of x can be found using the inverse of matrix A such that

General Form

Garin-Lowry Model

Ax y x

y Ix Ax

y (I A)x

(I A) 1 y x

The object is to find x given A and y . This is done by solving for x :

Matrix Operations in Excel

Select the cells in which the answer will appear

Matrix Multiplication in Excel

1) Enter “=mmult(“

2) Select the cells of the first matrix

3) Enter comma “,”

4) Select the cells of the second matrix

5) Enter “)”

Matrix Multiplication in Excel

Enter these three key strokes at the same time:

control

shift

enter

Matrix Inversion in Excel

Follow the same procedure Select cells in which answer is to be

displayed Enter the formula: =minverse( Select the cells containing the matrix to be

inverted Close parenthesis – type “)” Press three keys: Control, shift, enter