# Square Matrix

Topics: Matrix, Diagonal matrix, Matrices Pages: 22 (1347 words) Published: October 30, 2013
Matrix Algebra
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 + …+
j1
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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