MatricesMatrices For grade 1, undergraduate studentsFor grade 1, undergraduate students

Made by Department of Math. ,Anqing Teachers collegeMade by Department of Math. ,Anqing Teachers college

Definition. A rectangular array of numbers composed of m rows and n columns

is called an matrix (read m by nmatrix). We also say that the matrix A is of, orhas, size . 

11 12 1

21 22 2

1 2

...

...

...

n

n

m m mn

a a a

a a aA

a a a

m n

m n

1 Some notations

The elements

form the i-th row of A ,and the elements

form the j-th column of A. We will oftenwrite

for A.

1 2, ,i i ina a a

1 2, ,...,j j mja a a

( )ij m nA a

Definition. If

are matrices, then iff

for i=1,2…, m and j=1,…,n.

( ) ( )ij ijA a and B b

m n A B

ij ija b

Definition. If

are two matrices, their sum A+B , is the matrix

, where

i=1,2…, m , j=1,2…,n.

( ) ( )ij ijA a and B b

m n( )ijC c

,ij ij ijc a b

.Matrix opertions

Definition. If is an matrix and r is a number then rA, the scalar multiple of A by r, is the matrix

where

i=1,2…, m and j=1,…,n. 

( )ijA a m n

( )ijC c

,ij ijc ra

Proposition 1. The matrices of size form a vector space under the operations of matrix addition and scalar multiplication. We denote this vector space by Mmn.The dimension of the vector space Mmn is not hard to compute. We take our lead from the method we used to show that dim Rn=n. Introduce the matrix by the requirement

m n

m n ( )rs ijE e

0 , ,

1 , .ij

if i r j se

if i r j s

.Some properties

Proposition 2. The vectors

form a basis for Mmn . Thereforedim

EXAMPLE 1.

| 1, 2,..., , 1, 2,...,rsE r m s n

mnM mn

3 1 4 0 1 3

2 0 1 2 9 4

2 1 0 7 6 1

3 0 1 1 4 3 3 0 7

2 2 0 9 1 4 4 9 3 .

2 7 1 6 0 1 5 5 1

EXAMPLE 2.

1 4 6 0 1 0 1 1 4 7

2 0 1 7 9 1 2 3 7 9

1 0 4 1 6 1 0 4 1 7

2 1 0 2 1 3 7 7 9 9

1 5 5 4 8

1 2 4 14 18

2 Matrix products

Definition. If is an matrix and is an matrix, their matrix product is the matrix, where

( )ijA a m n( )ijB b n p

A B m p

( )ijAB c

1

n

ij ik kjk

c a b

1,..., , 1,..., .i m j p

Remark. Note that for the product of A and B to be defined the number of columns of A must be equal to the number of rows of B. Thus the order in which the product of A and B is taken is very important, for AB can be defined

without AB being defined.

EXAMPLE 4. Compute the matrix product

Solution. Note the answer is amatrix

4

(1 2 3) 5 .

6

4

(1 2 3) 5 (4 10 18) (32).

6

Remark. Note that the product

is not defined.

4

5 (1 2 3)

6

EXAMPLE 5. Compute the matrix product0 1 1 0 1 1

0 0 1 0 0 1 .

0 0 0 0 0 0

Answer .0 0 1

0 0 0

0 0 0

Definition. A matrix A is said to be a square matrix of size n iff it has n rows and n columns (that is the number of rows equals the number of columns equals n).

Remark. It is easy to see that if A and B are square matrices of size n then the products AB and BA are both defined. However they may not be equal.. 

EXAMPLE 7. Let1 0 3 0

0 3 2 1A and B

Compute the matrix products AB and BA. Solution. We have

1 0 3 0 3 0 3 0 1 0 3 0,

0 3 2 1 6 3 2 1 0 3 2 3AB BA

and so we see that AB BA.

Remark. As the preceding example

shows even if AB and BA are defined we should not expect that AB=BA.

Notation. If A is a square matrix then AA is defined and is denoted by A2.

Similarly,

is defined and denoted by .

...A A

n timesnA

EXAMPLE 8. Let0 0

.1 0

A

Calculate .2A

Solution. We have

2 0 0 0 0 0 0

1 0 1 0 0 0A

.The rules of matrix operations(1) A+B=B+A(2) A+(B+C)=A+(B+C)(3) r(A+B)=rA+rB(4) A+0=A (5) 0A=0 (6) A+(-1)A=0(7) (r+s)A=rA+sA(8) (A+B)·C=A·C+C·B(9) 0·A=0=A·0(10) A·(B·C)=(A·B) ·C

3 Special types of matrices

Diagonal matrices.

11 0

0 nn

a

A

a

Triangular matrices. A square matrix A is said to be lower triangular iff A= ( )ija

where 0ija if j i For example

1 0 0

0 2 0

3 1 3

is a lower triangular matrix.

The Zero matrix. The zero matrix is the matrix 0 all of those entries are 0.

Idempotent matrices. A square matrix A is said to be idempotent iff 2A A

Nilpotent matrices. A square matrix A is said to be nilpotent iff there is an integer q such . 0qA

Nonsingular matrices. A square matrix A is said to be invertible or nonsingular iff there exists a matrix B such thatAB=I and BA=I. Denoted by . 1A

For example if 0 1 0

1 0 0

0 0 1

A

then

1

0 1 0

1 0 0 .

0 0 1

A

A nilpotent matrix is not invertible. For suppose that A is a nilpotent matrix that is invertible. Let B be an inverse for A. Since A is nilpotent there is an integer q such that 0.qA

Then1 1 10 q q q qA B A AB A I A

so

1 0.qA

If we repeat this trick q-1 times we will get 0.A

But then

which is impossible.0 0,I AB B

Symmetric and skew-symmetric matrices. A square matrix A= is said to symmetric iff

for it is said to be skew-symmetric iff for

ijaij jia a

- ,ij jia a , 1, 2,..., .i j n

For example

0 1 2 31 0 1

1 4 5 60 0 0

2 5 7 81 1 3

3 6 8 9

and

are symmetric matrices, and

1 1 20 1

1 0 31 0

-2 3 0

and

are skew-symmetric matrices.

Proposition 3 A matrix2 2

a bA

c d

is nonsingular iff If then

0.ad bc 0,ad bc

1 1 d bA

c aad bc

PROOF. Suppose that Let

0.ad bc 1 d b

Bc aad bc

Then 1 d b a bBA

c a c dad bc

1

01

0

1 0.

0 0

da bc bd bd

ca ac ca adad bc

ad bc

ad bcad bc

I

and therefore A is nonsingular with

1 1.

d bA

c aad bc

Suppose conversely that A is nonsingular, but that . We will deduce a contradiction. Let0ad bc

.d b

Cc a

Then computing as above

0( ) 0.

0

ad bcCA ad bc I

ad bc

This gives the equation

1 1 1( ) ( ) 0 0.C CI C AA CA A A Therefore

0 0.

0 0

d bC

c a

So that

0, 0, 0, 0.a b c d But then A=0 also, so

1 10 0.I AA A So

and hence 1=0, which is impossible.

1 0 0 0

0 1 0 0

4 SOME EXERCISES

1. Perform the following matrix multiplications

0 1 0 1 0

1 1 0 0 1 ,

0 0 2 1 0

1 0

1 0 1 0 0 1 2 2,

0 1 0 1 1 0 2 2

0 1

0 0 0 1 2 3

1 0 0 4 5 6 .

0 0 0 7 8 9

2. Which of the following matrices are nonsingular, idempotent, nilpotent, symmetric, or skew-symmetric?

1 1 0 1, ,

0 0 1 0

1 1 1 0, ,

1 1 0 1

1 1 1 0, ,

1 1 1 0

1 1 4 0, .

1 1 0 2

A F

B G

C H

D J

3. If A is an idempotent square matrix show I-2A is invertible (Hint: Idempotent correspond to projections. Interpret I-2A as a reflection. Try the case first. Then try to generalize.)2 2

Thanks!!!