Matrices are sets of numbers or expressions arranged in rows and columns. A matrix is defined by its order, or the number of rows and columns it contains. There are several types of matrices including square, zero, identity, and triangular matrices. Operations on matrices include finding the transpose, determinant, and reducing a matrix to row echelon form. Determinants are values that can be calculated for square matrices and have various properties when operating on matrices.

1. MATRICES
Presented by:
Farhan Tariq
Ali Hassan
Zargham Abbas

2. WHAT IS A MATRIX ?
• A matrix is a set of elements arranged in rows and columns
or rectangular array of number enclosed in brackets.
2 3 7
1 -1 5
A=
B=
1 2
3 4
4 6
Both A and B are examples of matrices.
• Generally they are represented in parenthesis

3. ORDER OF MATRIX
• Number of rows and columns which a matrix has is called order of matrix.
Rows are written horizontally and columns are vertically.
1 5 8
2 6 8
3 7 5
A= Matrix A has 3 rows and 3 columns, so it is matrix of order
3x3.Rows are represented with “m” and columns are
with “n”.Generally, m x n is called order of matrix.
Similarly,
2 5
3 7
4 9
3 4 7 8
4 3 1 0
5 6 4 5
B= C=
B is a matrix of order 3x2 while C is a matrix of order 3x4.

4. TYPES OF MATRICES
• Square matrix
When m=n, a matrix is called square matrix. Example A=
where 2,5,7 are called Diagonal elements.
• Trace of any matrix is found by adding the elements in diagonal. Trace of A is 2+5+7=14.
• Equal matrices
Two matrices are said to be equal if each element of first matrix is equal to the corresponding
element of second matrix.
A= B= A and B are only said to be equal in this type of state.
2 3 7
2 5 8
3 6 7
2 3
3 4
2 3
3 4

5. • Zero Matrix
If every element of a matrix is zero, then it is called zero or null matrix. N=
• Upper triangular matrix is a matrix whose all elements below the diagonal elements are
zero.
F= 1 2 3
0 5 6
0 0 1
• Lower triangular matrix is a matrix whose all elements upper the diagonal elements are zero.
D= 1 0 0
2 5 0
3 6 7
0 0
0 0

6. • Identity Matrix
Both upper and lower triangular matrix whose all diagonal elements are I=
‘1’ is called identity matrix.
• Transpose of matrix
It is obtained by interchanging the rows and columns of a matrix. If A =
• Symmetric Matrix
A matrix A such that A=AT is called symmetric matrix.
If At =-A, then it is called a skew symmetric matrix.
1 0 0
0 1 0
0 0 1
2 9
3 4
2 3
9 4
Transpose of A=

7. • Orthogonal Matrix
A matrix A is called to be orthogonal only if AAT =ATA= Identity matrix.
We can say that orthogonal means AT=A-1 .
• Properties of matrices
1. (AB)-1=B-1A-1
2. (At)t =A and (kA)t=k At
3. (A+B)T=At + Bt
4. (AB)t=Bt At

8. DETERMINANTS
• Determinant of 2x2 matrix
Consider a matrix A = , Then its determinant can be founded as follows:
|A|= 2 3
|A|=2x6 – 4x3 =12 – 12= 0
• Determinant Of 3x3 Matrix
3 2 1
2 4 7 = 3 - 2 + 1 =3(12-14) – 2(6-7) + 1(4-4)= -4
1 2 3
2 3
4 6
4 6
4 7
2 3
2 7
1 3
2 4
1 2

9. • Properties of Determinants
• |At|=|A| , A matrix and its transpose has same determinant.
• |A|= 0 ,If two rows or columns are same.
• K |A|= k det A
• Det A= -|A|, if we interchange one or two rows and columns.
• |A|=0, If all elements of any row or column are zero.
• |AB|= |A||B|
• If matrix is upper or lower triangular, determinant can be found by multiplying the elements of
diagonals.

10. Echelon Form
• A matrix is in row echelon form when it satisfies the following conditions. The
first non-zero element in each row, called the leading entry, is 1. Each
leading entry is in a column to the right of the leading entry in the previous
row.
A =
1 2 3
0 0 4
0 0 0

11. • Reducing a matrix in row echelon form
To reduce any square matrix into row echelon form , utilize these steps in order to get your
desired row echelon form.
1st step
2ND step
1 0 0
0 1 0
0 0 1
3rd
step
4th step
4th
step
5th step
6th step