# Square Matrix

**Topics:**Matrix, Diagonal matrix, Matrices

**Pages:**22 (1347 words)

**Published:**October 30, 2013

http://elearning.usm.my

Md Harashid bin Haron, Ph.D.

Accounting Section,

School of Management,

Universiti Sains Malaysia (USM),

11800 Pulau Pinang, Malaysia

Email: harashid@usm.my ; mdharashid@gmail.com

Matrices?

A rectangular array of numbers consisting m

horizontal rows and n vertical columns.

5

3 4

2 2 1

6 4 2

A=

5

3 4

2 2 1

6 4 2

A has a size of 3 x 3; 3 x 3 matrix; 3 rows and 3

columns (row is specified first followed by column),

the numbers are called entries. For the entry of Aij,

the row subscript is i and the column subscript is j.

Two common methods to

denote entries

a b

c d

Or

a11 a12

a a

21 22

Row vs Column vector

3 x 4 y 5 z 0

2 x 2 y z 0

6 x 4 y 2 z 0

Numerical coefficients and their relative positions

(system of linear equations)

5

3 4

2 2 1

6 4 2

and

0

0

0

Matrix or matrices, uses brackets/parentheses,

represented by bold letters e.g., A, B, C, Z etc.

Equality of Matrices

A=[aij] and B=[bij] are equal if and only if

they have the same size, and aij=bij for

each i and j.

6 4 3

2.2

Eg.

2 =

3 2 3 2

Transpose of a matrix

AT , a matrix whose ith row transpose of an mxn

matrix A is the nxm is the ith column of A.

1 2

T 1 3

Eg. A=

=> A = 2 4

3 4

Special Matrices

•

•

•

•

•

•

Zero Matrix, Omxn

Square Matrix (when m=n) of order n.

In square matrix of order n, the a11, a22,

a33…ann entries are called main diagonal

A square matrix A is a diagonal matrix if all

the entries off the main diagonal are zeroes,

aij=0 for i≠j.

In is nxn identity matrix and is a diagonal

matrix whose main diagonal entries are 1’s.

A is a triangular matrix when either all entries

below the main diagonal entries are zero, aij=0

for i>j (upper triangular matrix) or all entries

above the main diagonal entries are zero, aij=0

for i

1 2

3 4

1 1 2 3

3 2 4 4

+ B =

=>

1 3

2 4

2 5

5 8

Example

A=

=>

1 2

3 4

1 3

+ (-B)=(-1)

2 4

1 1 2 3

3 2 4 4

=>

0 1

1 0

Scalar Multiplication

If A is an mxn matrix and k is a real

number then, by kA, we denote the mxn

matrix obtained by multiplying each entry

in A by k so that (kA)ij=kAij. This operation

is called scalar multiplication, and kA is

called a scalar multiple of A.

Properties of Scalar

Multiplication (pg 250)

1. k(A + B) = kB + kA

2. (k + l)A = kA + lA

3. k(lA) = (kl)A

4. 0A = O

5. kO = O

6. (A + B)T = AT + BT

7. (kA)T = kAT

Example

1 3

2

2 4

=>

2 6

4 8

Matrix Multiplication

Let A be an mxn matrix and B be an nxp

matrix. Then the product AB is mxp matrix

whose entry ABik is given by

n

(AB)ik = A ijB jk = Ai1B1k + Ai2B2k + …+

j1

AinBnk.

Properties of Matrix

Multiplication (pg 256)

1. A(BC) = (AB)C

(associative property)

2. A(B + C) = AB + AC (distributive property)

(A + B)C = AC + BC

3. kAB=k(AB)=(kA)B=A(kB)

4. (AB)T=BTAT

(ATBC)T = CTBT(AT)T = CTBTA

Note: IT=I

Example

1 2

A=

3 4

; AB

=>

=

B=

1 3

2 4

1 2 1 3

3 4 2 4

1.1 2.2 1.3 2.4

3.1 4.2 3.3 4.4

=

5 11

11 25

Note

If A is a square matrix and p is a positive integer,

then the pth power of A, written Ap, is the

product of p factors of A.

Ap =A.A….A (up to p factors)

If A is nxn, we define A0=In

Note: Ip = I

Matrix Equation

5 x1

3 4

2 2 1

x2

6 4 2 x

3

=>

3x1

2 x

1

6 x1

4 x2

2 x2

4 x2

=

1

2

3

5 x3

1x3

2 x3

=

1

2

3

3 x1...

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