A matrix is an ordered rectangular array of numbers. The size of a matrix is given by the numbers. The size of a matrix is given by the number of rows and the number of columns.Let m denote the number of rowsLet m denote the number of rowsLet n denote the number of columns.Let aij denote the entry in the ith row and the jth column.jth column.

nm

naaa

nm

11211

n

n

aaa

aaa

22221

11211

naaa 22221m rows

mnmm aaa 21 mnmm aaa 21

n columns

Zero matrixijij

square matrix square matrix

The Identity Matrix, denoted , is a diagonal matrix of order nxn with all the

nI

diagonal matrix of order nxn with all the diagonal entries equal to 1.

00

00

000

000

22

11

d

d

DO

00

00

000

000 22

d

dDO

mm

001

00000 dmm

matrixidentity an denote Let 010

001

nnII n

100

n

A square matrix all its elements below the main diagonal are zeros. 0

i ja

0

ijai j

2 3 3

0

0 0

7 1

9

A

0 0 9

A square matrix all its elements above the main diagonal are zeros. 0

i ja

0

ijai j

2 0 0

5 7

2 8 9

0A

2 8 9

The matrix A is 3x3 of the form

2

i>ji2

i>j

i jij

ia

i j

i ji j

2 3 411 12 13a a a

4 4 5

6 6 6

A11 12 13

21 22 23

a a a

A a a a

a a a6 6 6

31 32 33a a a

2

5

i i j

a i j

5ija i j

i j i j

3

i j i j

j i j

3ij

j i jb

i j i j

i j i j

If

A and B are 3x3 find A-2B

5 1 2a a a1 1 1 2 1 3

2 1 2 2 2 3

5 1 2

4 5 1

a a a

A a a a

3 1 3 2 3 3 6 6 5

3 3 4

a a a

3 3 4

3 6 5

3 6 9

B

3 6 9

1 7 1 0

2 2 7 1 1A B2 2 7 1 1

0 6 1 3

A B

Equality of Matrices:Two matrices are equal if they have the Two matrices are equal if they have the

same size and their corresponding entries are equal. are equal.

Find x and y that satisfies the following equationequation

x y y yx y y y

y

y=2y=2x+6=2 x=-4

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

ij ij ij

1,2,..., 1,2,...,and

ij ij ij

i m j n1,2,..., 1,2,...,and

i m j n

Note: Only matrices of the same dimension can Note: Only matrices of the same dimension can be added

Addition of Matrices

nn bbbaaa 1121111211

n

n

n

n

bbb

bbb

Baaa

aaa

A 22221

11211

22221

11211

and

mnmmmnmm bbbaaa 2121

nn

bababa

bababa 1112121111

nn

bababa

bababaBA 2222222121

mnmnmmmm bababa 2211

A

B

3

4

1

2

4

6

CA

B

5 6

3 4

8 10

C

Multiplying a matrix by a real number (scalar) results Multiplying a matrix by a real number (scalar) results in a matrix with each entry multiplied by the scalar. Let k be a real number.

naaa 11211 nkakaka 11211

k be a real number.

n

n

aaa

aaa

A 22221

11211

n

n

kakaka

kakaka

kA 22221

11211

mnmm aaa 21 mnmm kakaka 21

k(A + B) = kA + k B(x+y) A = x A + y A(x+y) A = x A + y A(x y) A = x ( y A)o A = Ox O = Ox O = O

C = A - BC = A - Bis defined byC = A + (-1) BC = A + (-1) B

nn bababa 1112121111

nn

nn

bababa

bababa

BA 2222222121

1112121111

bababa

BA

mnmnmmmm bababa 2211

The scalar matrix can be written as .

C = InC = In

0 0 0x 0 0 0

0 0 0

x

x0 0 0

0 0 0

x

x0 0 0

0 0 0

x

x

Matrices A and B have these dimensions:

Matrices A and B can be multiplied if:

The resulting matrix will have the dimensions:

pn

bbb

bbb

aaa

aaa 1121111211

pn

bbb

bbbB

aaa

aaaA

2222122221 and

npnnmnmm bbbaaa 2121

ccc

22221

11211

p

p

ccc

ccc

CAB

Entry cij is obtained by taking the sum of the products of the entries of

21 mpmm ccc

CABproducts of the entries of the ith row in A with the jth column in B.

122122112121

112112111111

...

...Entry

nn

nn

bababac

bababacjth column in B.

212212121122

122122112121

...

...

nn

nn

bababac

bababac

mxn mxs sxnmxn mxs sxn

s

1

s

ij ik kjk

c a b1

1, 2,...,k

i m

1, 2,...,j n

A

2

3

1

1

and

B

1 1 1 A

1

1

1

0

and

B

1 0 21

0[3 x 2] [2 x 3]

A and B can be multiplied

[3

x 3][3

x 3]

2*1 3*0 2

5 2 82*1 3*1 5 2*1 3*2 8

2*1 3*0 2

5 2 8

1*1 1*1 2 1*1 1*2 3 2 1 3

2*1 3*1 5 2*1 3*2 8

1*1 1*0 1C

1*1 0*0 1 1 11*1 0*1 1 1* 0 11 *2 1

2x3 3x2 2x2

2 31 1 1

1 1

B x A 1 11 0 2

1 0

B x A

4 44 4

4 3 =

Properties of Matrix Multiplication

1.AB BA

AB C A BC

2-

3. )

AB C A BC

A B C AB AC

2-

(

4. A m n I I

(B+C)A=BA+CA

If is an matrix and and are4. m nA m n I I

m m n n

If is an matrix and and are

and identity matrices, respectively,

I A AI A

then

m nI A AI A

5- Amxn 0nxk =0mxk

0kxm Amxn = 0kxn

If A is an m x n matrix with elements aij, then the transpose of A, denoted, AT, is an n x m matrix with elements aji.

aaa aaa

n

n

aaa

aaa

A 22221

11211

m

m

T aaa

aaa

A 22212

12111

mnmm aaa

A

21 mnnn

T

aaa

A

21mnmm aaa 21 mnnn aaa 21

TT TT

T T T

T T T

T T

A square matrix B is said to be symmetric if

B = BTB = BT

brs = bsr s rr=1,2, ,ms=1,2, ,ns=1,2, ,n

Any diagonal matrix is symmetric

A squared zero matrix is symmetric

The transpose of an upper triangular matrix is The transpose of an upper triangular matrix is a lower triangular matrix

A squared matrix is said to skew-symmetric ifA = - AT

0 1 2

1 0 3A 1 0 3

2 3 0

A

0 1 2

1 0 3TA 1 0 3

2 3 0

TA

TA A

The common term of a skew-symmetric matrix can be written as

r sars

sr

r sa

a r s

If the common element of a 3x3 matrix is given by

rs

Show that A is a skew-symmetric matrixShow that A is a skew-symmetric matrix

0 1 10 1 1

1 0 1A 1 0 1

1 1 0

A

0 1 10 1 1

1 0 1TA

1 1 0TA A TA A

X =(x1,x2, .,xn) 1xnAT =(1,1, .,1) 1xnAT =(1,1, .,1) 1xn

xi = X A = AT XT

xi2 = X XTxi = X X

A square matrix is said to be orthogonal ifT TT T

T H E I D E N T I T Y M AT R I X I S O R T H O G O N AL AN D S YM M E T R I CO R T H O G O N AL AN D S YM M E T R I C

- 1 - 1- 1 - 1

2 2A = 2 2A =1 - 1

2 22 2

- 1 - 11 - 1 - 11A =

1 - 12

T - 1 11A =

- 1 - 12 - 1 - 12

T TT T2

If A is a square matrix and k is a positive integer, the power k of A is defined asinteger, the power k of A is defined asAk = A . A ..A

So, A2 = A . ASo, A = A . Aand A0 = I

A square matrix A is said to be idempotent if

Ak = A for any positive integer k

The identity matrix is idempotentThe identity matrix is idempotentThe square zero matrix is idempotent

Show that the matrix A defined below is idempotent 1 1

2 21 1

A1 1

2 2

1 1 1 1 1 1

A

2

1 1 1 1 1 1

2 2 2 2 2 21 1 1 1 1 1

A A A A1 1 1 1 1 1

2 2 2 2 2 2

3 2A A A A A A

Assume Ak-1 = AAk = Ak-1 A =A A=AAk = Ak-1 A =A A=ASo Ak = A for any positive integer ki.e. A is idempotent

Definition: Let A be an n n matrix. An inverse of A Definition: Let A be an n n matrix. An inverse of A is an n n matrix B such that:

AB = In and BA = InAB = In and BA = In

A is then called invertible.A is then called invertible.B is called an inverse of AIf A has no inverse, it is called singular.If A has no inverse, it is called singular..

The inverse (if exists) is unique.The inverse (if exists) is unique.Proof:Assume A is invertible, with two inverses B and C, i.e.Assume A is invertible, with two inverses B and C, i.e.A B = B A = IandandAC = CA = I,

(BA)C=B(AC)=BI=B(BA)C=IC=C(BA)C=IC=C

Thus, B=C Thus, B=C

Since the inverse ,if it exists, is unique, we call it A-1call it A

A A-1 = A-1 A = IA A-1 = A-1 A = In

Let A and B be invertible matrices of the same size, and k be a nonzero scalar. Then:and k be a nonzero scalar. Then:

1. I 1 = I

(A 1) 1 = A2. (A 1) 1 = A

3. (kA) 1 = k 1A 1

4- (AB) 1 = B 1A 1

5- (An) 1 = (A 1)n5- (A ) = (A )

(AT) 1 = (A 1)T6- (AT) 1 = (A 1)T

7- If A is orthogonal then it is invertible andA-1=ATA =A

If A and B are nxn invertible matrices, prove that AB is invertible also.that AB is invertible also.

(AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1(AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1

n-1 -1

n(AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1(AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1

n-1 -1

nn n

Find the inverse of the 2 2 matrix:

1

1

1

1

If3 1

5 2A

Show that

5 2

Show that A2 -5A+I2=0Using this result find A-1Using this result find A-1

A2 - 5A + I2 =0A-1 A A 5

A-1 A + A-1 = 0A-1 A A 5

A-1 A + A-1 = 0A 5 I + A-1 = 0A-1 = 5 I A

-1

3 53 5

2 3 A =If

4 71 4 7

2 4B

a m atrix X su ch th atFind

a m atrix X su ch th at

X B =A

Find

X B = AX B B-1 = A B-1X B B-1 = A B-1

X I = A B-1

Let A, B, and X be 3

invertible matrices of Let A, B, and X be 3

invertible matrices of the same size. Solve the following matrix the same size. Solve the following matrix equation for X:

(A 1XB) 1

= (BA)2(A 1XB) 1

= (BA)2

Note: Be careful with the order of the matrix Note: Be careful with the order of the matrix multiplication.

Answer: X = (B2AB) 1

= B 1A 1(B 1)2Answer: X = (B2AB) 1

= B 1A 1(B 1)2

(A-1 X B)-1 = (B A)2

B-1 X-1 (A-1 )-1 = (B A)2B-1 X-1 (A-1 )-1 = (B A)2

B-1 X-1 A = (BA)2

B B-1 X-1 A = B (BA)2

I X-1 A A-1 = B (BA)2 A-1I X A A = B (BA) AX-1 I = B BA BA A-1

X-1 = B2 A BX-1 = B2 A BX = ( B2 A B)-1X = ( B A B)

Show that the inverse of the general 2 2

matrix:Show that the inverse of the general 2 2

matrix:

baA

dc

baA

dc

bdA 1

ac

bd

bcadA 1

1

1

- 1 x yA s s u m e A =

w z

- 1

A s s u m e A =w z

a b x y 1 0A A = =

c d w z 0 1A A = =

c d w z 0 1

a x + b w = 1

c x + d w = 0c x + d w = 0

- c a x - c b w = - c

c a x + a d w = 0c a x + a d w = 0

- cw = a d - b c 0

a d - b cd

x = a d - b c 0a d b c