Definations and examples of Matrices

2. MATRICES
PRESENTED BY: MISBAH MEHMOOD
ROLL No: 2923
PRESENTED TO: MR WAJAHAT

3. Contents:
 Matrix
 Order of matrix
 Diagonal matrix
 Zero matrix
 Square matrix
 Identity matrix
 Rectangular matrix
 Transpose of matrix
 Symmetric matrix
 Skew symmetric matrix
 Echelon form of matrix
 Reduce echelon form of matrix
 Rank of matrix
 Hermition matrix
 Skew hermition matrix

4. Matrix:
Rectangular array of number enclosed by a pair of bracket is
known as matrix . The number inside the matrix is known as
Entries of matrix.
For example:

5. Order of matrix:
Number of rows multiply number of column is known as order
of matrix.
Represented by: m x n
Example:

6. Identity matrix:
Square matrix having 1’s on the main diagonal and 0’s on the
rest of the matrix.
For example:

7. Null matrix:
The square or rectangular matrix whose each element is zero.
For example:
Rectangular matrix:
The matrix in which number of rows are not equal to number
Of column
For example:

9. Transpose of matrix:
Matrix obtained by changing rows and column.
If order of matrix is m*n then n*m is transpose of matrix.
For example:

10. Symmetric matrix:
If for a square matrix A=[aij], 𝐴 𝑡
=A , then A is called symmetric
Matrix .
In a symmetric matrix aij=aji for each pair (i,j).
Represented as :
𝐴 𝑇
= 𝐴
For example:

11. Skew symmetric matrix:
for a square matrix A=[aij], 𝐴 𝑇 = −𝐴, 𝑡ℎ𝑒𝑛 𝑎 𝑖𝑠 𝑠𝑘𝑒𝑤 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐
Matrix
Represented as:
𝐴 𝑇
= −𝐴
For example:

12. Hermition matrix:
A square matrix A=[aij]n*n with complex entries is hermition
Matrix
Represented as:
For example:

13. Skew hermition matrix:
A square matrix with complex entries is skew hermition
Matrix
Represented as:
For example:

14. Echelon form of matrix:
Matrix in echelon form satisfied the following conditions:
 The first non zero element in row is leading entry.
 All the entries below the leading entry are zeros.
 Row with all zero elements if any are below row having
non zero element.
For example:

16. Reduce echelon form of matrix:
A matrix is in a reduce echelon form when it satisfied the
following condition:
 Leading entry of each row in non zero row is 1.
 Each leading 1 is the only non zero entry in its column.
for example: