Square Matrix

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: [e-mail] ; [e-mail]

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