REVIEW OF
MATRICES
MATRIX
• A matrix is simply a set of numbers arranged in a rectangular
table. We usually write matrices inside parentheses ( ) or
brackets [ ].
• Matrix is an array of mn values with m rows and n columns.
• Amxn (indicating the size of a matrix)
• If aij is an entry of the matrix then if it is in the ith row and jth
column
• 1 ≤ ⅈ ≤ 𝑚 ,1 ≤ 𝑗 ≤ 𝑛
A = 𝑎𝑖𝑗
Matrix Notation
– A matrix is written with ( ) or [ ] brackets.
– Do not confuse a matrix with a determinant which uses vertical bars | |. A
matrix is a pattern of numbers; a determinant gives us a single number.
– The ith row of matrix A is
– The jth column of matrix A is
ai1, ai2, ……., ain
a1 j
a2 j

amj
 a11 a12 a13  a1n 
a

a
a

a
23
2n 
 21 22
A= 


 




 
 
am1 am 2 am3  amn 
ORDER OF A MATRIX
A matrix which has m rows and n columns is called a matrix of order m x n
eg. The order of
5
3
−1 4
matrix is 2x3
9 −2
Note:
(a) The matrix is just an arrangement of certain quantities.
(b) The elements of a matrix may be real or complex numbers. If all
the elements of a matrix are real, then the matrix is called a real matrix.
(c) An m x n matrix has m.n elements.
Elements in a matrix
• The elements in a matrix A are denoted by aij, where i is the row
number and j is the column number.
• Example 1
• Consider the matrix
The element a21 = 1, since the element in the 2nd row and 1st column is 1.
• The element a13 = 9, since the element in the 1st row and 3rd column is 9.
CLASSIFICATION OF
MATRICES
1. Square matrix is a matrix with equal rows and columns.
An x n
 a11 a12  a1n 
a

a

a
2n 
=  21 22
 

 


 an1 an 2  ann 
Main diagonal of A has the entries a11, a22, …,ann
2. Column Vector or row vector
-Column vector or column matrix is a matrix with only one column
and m rows (m x 1)
− 3
4
A= 
8
 
0
-Row vector or row matrix is a matrix with only 1 row and n
columns (1xn).
B = [ -1 5 -7]
3.Diagonal Matrix is an n x n matrix such that aij = 0 for i ≠ j
0
0 
a11 0
0 a

0
0
22

A=
 
   


0
0 ann 
0
4. Zero/Null matrix is a matrix all of whose entries are zeroes.
0 0 
B=

0
0


5. Scalar matrix is a diagonal matrix where all entries along the principal
diagonal are all equal, that is, a 11= a 22= a nn.
4 0 0
A = 0 4 0
0 0 4
6. Identity matrix or unit matrix, is scalar matrix whose elements along the
diagonal are all equal to one: aii = 1; 1 ≤ i ≤ n
In
1
=
0


0
0
1
0
0
0

1

7.
Triangular Matrices
Upper Triangular Matrix is an n x n matrix such that aij = 0 for i > j. (all
elements below the diagonal are equal to zero)
a11 a12 a13 
C3 x3 =  0 a22 a23 
 0
0 a33 
Lower Triangular Matrix is an n x n matrix such that aij = 0 for i < j. ( all
elements above the diagonal are equal to zero)
0 
 a11 0
D = a21 a22 0 
a31 a32 a33 
8. Coefficient Matrix
 a11
a
 21
 

am1
a12
a22

am 2
a1n 
 a2 n 
 

 amn 

Given the following systems of linear equations:
a11x1 + a12 x2 +  + a1n xn = b1
a21x1 + a22 x2 +  + a2 n xn = b2




am1 x1 + am 2 x2 +  + amn xn = bm
9. Augmented Matrix
[C|B]
 a11 a12  a1n
a
a

a
21
22
2n

 



am1 am 2  amn
b1 

b2 


bm 
10. Transpose of a matrix
If A is an m x n matrix then the n x m matrix B such that bij = aji is the
transpose of A and B = AT
A2 x 3
1
=
4
B = A3Tx 2
3

5 6
1 4 


= 2 5
3 6
2
PROPERTIES OF AT
T
T
1.(A )
=A
T
T
T
2.(A + B) = A + B
T
T
T
3.(AB) = B A
T
T
4.(rA) = rA
11. Symmetric Matrix is an n x n matrix such that A = AT
1

A = 2
3
2
4
6
3

6
5
12. Skew Symmetric matrix is an n x n matrix such A = − AT
2
3
 0


B = − 2
0
4
 − 3 − 4 0
0 − 2 − 3


T
B = 2
0
− 4
3
4
0 
13. Inverse of an n x n matrix A is the n x n matrix A-1 such that AA-1 = In
1 2
A=

3 4
−
2
1


−1
A = 3

1
−
 2
2 
14. Invertible matrix or nonsingular matrix is a matrix with an inverse
otherwise it is noninvertible or singular.
15. Equal matrices have identical corresponding elements
A and B are m x n matrices such that aij = bij
If
then x = 2, y = 7 and a = 3.
16. Orthogonal matrix
Let A be a square matrix, if the transpose of the matrix A is equal
to its inverse that is A-1 = AT , then A is called orthogonal
TRACE OF A MATRIX
SQUARE MATRIX
Diagonal Matrix
Atleast one 𝑎𝑖𝑗 ≠0 and
𝑎𝑖𝑗 =0 if ⅈ ≠ 𝑗
Lower Triangular
Upper Triangular
Lower Triangular
if aij = 0  i  j
if aij = 0  i  j
X
A =  0
 0
X
A =  X
 X
X
X
0
X
X 
X 
0
X
X
0
0 
X 
d1
0

 0
Scalar Matrix
0
d2
0
0
0 
d 3 
Unit Matrix
if d1 = d 2 = d 3 =  = a  0
if d1 = d 2 = d 3 =  = 1
a 0 0 
0 a 0


 0 0 a 
1 0 0
0 1 0 


0 0 1