The document discusses different types of matrices including identity matrices, inverse matrices, transpose matrices, symmetric matrices, orthogonal matrices, upper triangular matrices, lower triangular matrices, and diagonal matrices. It provides examples and properties of each matrix type. Key points covered include that identity matrices satisfy AI = IA = A, inverse matrices satisfy AB = BA = I, the transpose of a matrix is formed by interchanging rows and columns, and orthogonal matrices satisfy AAT = ATA = I.

1. 1. TYPES OF MATRICES
PRESENTED BY:
NAME: RANJAN GUPTA
ROLL NO. 221563`
BRANCH : BTECH ( CIVIL )

2. 1. Types of matrices
Identity matrix
The inverse of a matrix
The transpose of a matrix
Symmetric matrix
Orthogonal matrix

3. A square matrix whose elements aij = 0, for
i > j is called upper triangular, i.e., 11 12 1
22 2
0
0 0
 
 
 
 
 
 
n
n
nn
a a a
a a
a
A square matrix whose elements aij = 0, for
i < j is called lower triangular, i.e., 11
21 22
1 2
0 0
0
 
 
 
 
 
 
n n nn
a
a a
a a a
Identity matrix
1. Types of matrices

4. 4
Both upper and lower triangular, i.e., aij = 0, for
i  j , i.e., 11
22
0 0
0 0
0 0
 
 
 

 
 
 
nn
a
a
D
a
11 22
diag[ , ,..., ]
 nn
D a a a
Identity matrix
1. Types of matrices
is called a diagonal matrix, simply

5. In particular, a11 = a22 = … = ann = 1, the
matrix is called identity matrix.
Properties: AI = IA = A
Examples of identity matrices: and
1 0
0 1
 
 
 
1 0 0
0 1 0
0 0 1
 
 
 
 
 
Identity matrix
1. Types of matrices

6. If matrices A and B such that AB = BA = I,
then B is called the inverse of A (symbol: A-1);
and A is called the inverse of B (symbol: B-1).
The inverse of a matrix
6 2 3
1 1 0
1 0 1
B
 
 
 
 
 
 

 
Show B is the the inverse of matrix A.
1 2 3
1 3 3
1 2 4
A
 
 
  
 
 
Example:
1 0 0
0 1 0
0 0 1
AB BA
 
 
   
 
 
Ans: Note that
1. Types of matrices

7. The transpose of a matrix
The matrix obtained by interchanging the
rows and columns of a matrix A is called the
transpose of A (write AT).
Example:
The transpose of A is
1 2 3
4 5 6
 
  
 
A
1 4
2 5
3 6
 
 
  
 
 
T
A
For a matrix A = [aij], its transpose AT = [bij],
where bij = aji.
1. Types of matrices

8. Symmetric matrix
A matrix A such that AT = A is called symmetric,
i.e., aji = aij for all i and j.
A + AT must be symmetric. Why?
Example: is symmetric.
1 2 3
2 4 5
3 5 6
 
 
 
 
 

 
A
A matrix A such that AT = -A is called skew-
symmetric, i.e., aji = -aij for all i and j.
A - AT must be skew-symmetric. Why?
1. Types of matrices

9. Orthogonal matrix
A matrix A is called orthogonal if AAT = ATA = I,
i.e., AT = A-1
Example: prove that is
orthogonal.
1/ 3 1/ 6 1/ 2
1/ 3 2/ 6 0
1/ 3 1/ 6 1/ 2
 

 
 
 
 
 
 
A
1. Types of matrices
Since, . Hence, AAT = ATA = I.
1/ 3 1/ 3 1/ 3
1/ 6 2/ 6 1/ 6
1/ 2 0 1/ 2
T
A
 
 
 
 
 

 
 