Matrix Types of Matrix Scalar Multiplication Addition

2. A matrix is a collection of numbers arranged into a fixed
number of rows and columns. A matrix is defined as an ordered
rectangular of numbers. They can be used to represent systems
of linear equations.
Matrix Dimensions: The numbers of rows and columns of a
matrix are called its dimensions. Here is a matrix with three
rows and two columns:
a2 a2
a3 a3
a9 a0
A =
3x2
In above example, Capital A is the Matrix name and small “a” is an element name and
value of those elements or matrix is subscript of “a”.

3. There are several types of matrices, but the most commonly
used are:
 Rows Matrix
 Columns Matrix
 Rectangular Matrix
 Square Matrix
 Diagonal Matrix
 Scalar Matrix
 Identity Matrix
 Triangular Matrix
 Null or Zero Matrix
 Transpose of a Matrix

4.  Rows Matrix
A matrix is said to be a row matrix if it has only one row.
A=[123]
 Columns Matrix
A matrix is said to be a column matrix if it has only one column.
B =
 Rectangular Matrix
A matrix is said to be rectangular if the number of rows is not
equal to the number of columns.
A=
1
2
3
1 2 8
5 6 9

5.  Square Matrix
A matrix is said to be square if the number of rows is equal to the
number of columns.
A=
 Diagonal Matrix
A square matrix is said to be diagonal if at least one element of
principal diagonal is non-zero and all the other elements are
zero.
A=
 Scalar Matrix
A diagonal matrix is said to be scalar if all of its diagonal elements
are the same.
A=
1 2 3
6 7 8
9 6 5
1 0 0
0 2 0
0 0 3
2 0 0
0 2 0
0 0 2

6.  Identity Matrix
A diagonal matrix is said to be identity if all of its diagonal
elements are equal to one, denoted by I.
 Triangular Matrix
A square matrix is said to be triangular if all of its elements
below the principal diagonal are zero (lower triangular matrix) or
all of its elements above the principal diagonal are zero (upper
triangular matrix).
(Lower Triangular Matrix) (Upper Triangular Matrix)
1 0 0
0 1 0
0 0 1
I =
A =
1 0 0
2 4 0
3 5 6
1 2 3
0 4 5
0 0 6
A =

7.  Null or Zero Matrix
A matrix is said to be a null or zero matrix if all of its elements
are equal to zero. It is denoted by O.
,
 Transpose of a Matrix
Suppose A is a given matrix, then the matrix obtained by
interchanging its rows into columns is called the transpose
of A. It is denoted by At
then
0 0 0
0 0 0
0 0 0
O =
0 0 0
0 0 0O =
1 2 3
8 4 6A =
1 8
2 4
3 6
At =

8. • Multiplication of a Matrix by a Scalar
A matrix can be multiplied by a scalar (by a real number) as
follows:
To multiply a matrix by a scalar, multiply each element of the
matrix by the scalar. Here is an example of this. (In this example,
the variable a is a scalar.)
1 2 3
8 4 6a =
1a 2a 3a
8a 4a 6a

9. • Matrix Addition
If two matrices have the same number of rows and same number
of columns, then the matrix sum can be computed: If A is an MxN
matrix, and B is also an MxN matrix, then their sum is an MxN
Matrix formed by adding corresponding elements of A and B
Here is an example of this:
1 2 3
8 4 6
+
1 2 3
8 4 6 =
2 4 6
16 8 12

10. The determinant is a value that can be computed
from the elements of a square matrix.
The determinant of a matrix A is denoted by det(A)
or |A|. It can be viewed as the scaling factor of
the transformation described by the matrix.
In the case of a 2 × 2 matrix the specific formula
for the determinant is:
A =
a b
c d = ad-bc

11. In the case of a 3 X 3 matrix the specific formula
for the determinant is:
A =
a b c
d e f
g h i
= a e f
h i
- b d f
g i
+ c d e
g h
= aei + bfg + cdh – ceg – bdi – afh.

12. For a square matrix A, the inverse is written A-1
When A is multiplied by A-1 the result is
the identity matrix I. Non-square matrices do not
have inverses.
Note: Not all square matrices have inverses. A
square matrix which has an inverse is
called invertible or nonsingular, and a square
matrix without an inverse is called noninvertible or
singular.

14. Here's a C program to enter values in a matrix and print values in a matrix using For
Loops and Nested Loops with output and explanation.
# include <stdio.h>
# include <conio.h>
void main()
{
int mat[10][10] ; int i, j, row, col ;
clrscr() ;
printf("Enter the order of the matrix : ") ;
scanf("%d %d", &row, &col) ;
printf("nEnter the elements of the matrix : nn");
for(i = 0 ; i < row ; i++)
for(j = 0 ; j < col ; j++)
scanf("%d", &mat[i][j]) ;
printf("nnThe elements in the matrix are: nn");
for(i = 0 ; i < row ; i++)
{
for(j = 0 ; j < col ; j++)
{
printf("%d", mat[i][j]) ;
printf("t"); } printf("n");
}
getch() ;
}
Output of above program -
Enter the order of the matrix : 3 3
Enter the elements of the matrix :
1
2
3
4
5
6
7
8
9
The elements in the matrix are:
1 2 3
4 5 6
7 8 9