Matrix algebra

An introduction to Matrix Algebra

Algebra

MATRIXA matrix is an ordered rectangular array of numbers, arranged in rows and columns.

columns

rows

ORDER OF A MATRIX

The size or order of a matrix is described by its number of rows and the number of columns.

If a matrix, A, has m rows and n columns then A is described as an mxn matrix.

The numbers in a matrix are called its elements. The element in the ith row and jth column of a matrix is generally denoted by aij. A matrix with m rows and n columns is written or .

Row Matrix

A matrix with just one row is called a row matrix (or row vector).

jn aaaaA , 2 1 (1 x n)

Column Matrix

i

m

a

a

a

a

A 2

1

A matrix with just one column is called a column matrix.

(m x 1)

Matrices of the same order

Two matrices which have the Same number of rows and columns are said to be matrices of the same order.

Equal MatricesTwo matrices A = (aij) and B = (bij) are said to be equal if, and only if, each element aij of A is equal to the corresponding element bij of B.

In symbolic form this reads:

From this it follows that equal matrices are of the same order but matrices of the same order are not necessarily equal.

A=B aij = bij for all i and j

Null matrixAny matrix, all of whose elements are zero, is called a null or zero matrix and is denoted by O.

Matrix Addition

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

ijijij bac

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

Multiplication by a scalar

If A is a given matrix and a scalar then

A is the matrix each of whose elements is

times the corresponding element of A.

Thus A

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.

Equal Matrices

Two matrices A and B are said to be equal if, and only if, each element aij of A is equal to the corresponding element bij of B.

The Null matrix

Any matrix all of whose elements are zero is called a null or zero matrix

Transpose Matrix

A'

a11 a21 ,, am1

a12 a22 ,, am 2

a1n a2n ,, amn

Rows become columns and columns become rows

Square Matrix

B

5 4 7

3 6 1

2 1 3

Same number of rows and columns

Matrix Subtraction

C = A - BIs defined by

Cij Aij Bij

Let A and B be two matrices. If the number of columns in A is equal to the number of rows in B we say that A and B are conformable for the matrix product AB.

If A is order m×n and B is of order n×p, then the product AB is defined and is a matrix of order m×p.

Matrix Multiplication

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:

[m x n] and [n x p]

n = n

Matrix Multiplication

The resulting matrix will have the dimensions:

[m x n] and [n x p]

m x p

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

Note:

If A is an m×n and B is n×p matrix, then AB is an m×p matrix. Hence we see that BA is defined only when p=m.

Inversion

The Inverse of a Matrix

Let A be a square matrix. A matrix B such that AB=I=BA is called the inverse matrix of A and is denoted by A-1.

So if A-1 exists, we have AA-1=I=A-1A and the matrix is said to be invertible.

If a matrix has no inverse, then it is said to be non-invertible.

Definition:

The Inverse of a Matrix

IAAAA 11

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