The document defines and provides examples of different types of matrices, including: - Square matrices, where the number of rows equals the number of columns. - Rectangular matrices, where the number of rows does not equal the number of columns. - Row matrices, with only one row. - Column matrices, with only one column. - Null or zero matrices, with all elements equal to zero. - Diagonal matrices, with all elements equal to zero except those on the main diagonal. The document also discusses transpose, adjoint, and addition of matrices.

1. Mathematics.
Matrices.
Awais Bakshy.

2. Matrices.
 Introduction:
1. Joseph James Sylvester was a English mathematician (1814-1897) used the word matrix first
time.
 Matrix.
 Matrix is singular while matrices is plural.
 Definition:
 A matrix is a rectangular array of number, symbols or expressions arranged in rows and
columns.
 Fro Examples.
A=
2 3 5
6 7 9
11 8 11
C= 9 D=
2 9
6 8
E=
5
7
9

3. Matrices.
 Definition:
 A Matrix is a bracket containing an array of numbers or letters arranged in rows and in columns.
 For Examples.
A=
5 7 6
6 9 7
4 2 4
C=
5 6 3
7 1 6
4 6 8
 Elements Or Entries Of Matrix.
 Each numbers or letters used in a matrix is called elements or entries of matrix.
 Order Of Matrix.
 The Number of rows by the number of columns is called order of matrix.
 It is written as m × n ( Here it is not product that you multiply here this symbol means m by n)

4.  m n is called order of the matrix.
 We usually use capital latters such as A, B, C, D, etc. to represent the matrices and Small letters such
as a, b, c, d, etc. to indicate the entries of the matrices.
 Note:
 The elements(entries) of matrices need not always be numbers but in the study of matrices, we shall
take the elements of the matrices from Real numbers or complex numbers.
 Kinds Of Matrix.
 Row Matrix.
 A Matrix having only one row is called row matrix.
 A matrix having only one row with two or more columns is known as row matrix.
 A= 2 4 5 6 8 is a row matrix of order 1 × 5
Matrices.

5. Matrices.
 Column Matrix.
 A Matrix having only one column is called column matrix.
 A matrix having only one column with two or more rows is known as column matrix.
4
B= 6 is a column matrix of order 1 × 3
8
 Rectangular Matrix.
 If in a matrix the number of rows and the number of columns are not equal then the matrix is called
rectangular matrix.
 If m ≠ n, (If m is not equals to n), then the matrix is called a rectangular matrix of order m × n. that
is, the matrix in which the number of rows is not equal to the number of columns is said to be
rectangular matrix.

6. Matrices.
 For Example.
3 6
C= 9 11 is a rectangular matrix of order 3 × 2
4 2
 Square Matrix.
 If matrix in which number of rows and number of columns are equal then the matrix called square
matrix.
 If m=n, then the matrix of order m × n is said to be square matrix of order n or m. i.e., the matrix
which has the same number of rows and columns is called a square matrix.
 For Example.
4 6 3
A= 5 4 6 is a square matrix of order 3 × 3 . C= 3 is a square matrix.
3 4 2

7. Matrices.
 Null Matrix Or Zero Matrix.
 A matrix whose each element is zero is called a null or zero matrix.
 it is denoted by “0”.
 For Example.
A=
0 0
0 0
is null matrix of order 0.
 Diagonal Matrix.
 A square matrix in which all the elements except at least one element of diagonal are zero.
 A square matrix in which all its elements are zero except the diagonal which runs from upper left to lower
right is known as diagonal matrix.
 For Example.
A=
4 0
0 4
B=
1 0 0
0 1 0
0 0 1
C=
2 0 0
0 4 0
0 0 2

8. Matrices.
 Scalar Matrix.
 A diagonal matrix having equal elements in its diagonal is called scalar matrix.
 For Example. 2 0 0
A= 0 2 0 C=
4 0
0 4
0 0 2
 Negative Matrix.
 If signs of all the entries of a matrix A changed the new matrix obtained will be negative matrix of
matrix A is called negative matrix.
 It is denoted by –A.
 For Example.
-5 4 -3 5 -4 3
A= 6 8 1 -A= -6 -8 -1 is a negative matrix of A matrix.
3 9 5 -3 -9 -5

9. Matrices.
 Unit Matrix or Identity Matrix.
 A scalar matrix having each elements of a diagonal equal to 1 is called Unit Matrix.
 It is Denoted By Capital latter I.
 For Example.
1 0 0
A= 0 1 0 C=
1 0
0 1
0 0 1
 Transpose Of a Matrix.
 If A is a matrix of order m × n then a matrix of order n × m obtained by interchanging the rows and
columns of A matrix is called Transpose of A Matrix.
 It is denoted by 𝐴𝑡
.

10. Matrices.
 For Example.
3 4 2 3 4 7
A= 4 3 1 𝐴𝑡
= 4 3 2
7 2 8 2 1 8
 Adjoint of a Matrix.
 Let A=
𝑎 𝑏
𝑐 𝑑
Is a matrix.
 Then the matrix obtained by interchanging the elements of primary diagonal, i.e. a and d and by
changing the sings of the other elements, i.e. b and c.
 It is denoted by adj A.
adj A=
𝑑 −𝑏
−𝑐 𝑎

11. Matrices.
 Primary and Secondary Diagonal of a matrix.
A=
1 0
0 1
1
1
is primary diagonal. 0
0 is secondary diagonal.
 Addition of Matrices.
 If P and Q are two matrices of the same order then the sum is obtained by adding their corresponding
elements. The sum of P and Q is denoted by P+Q and its order will be equal to the order of matrix P and Q.
 For Example.
Let. P and Q
P=
3 2
5 1
Q=
1 2
3 4
Solution.
P+Q=
3 2
5 1
+
1 2
3 4

12. Matrices.
P+Q=
3 + 1 2 + 2
5 + 3 1 + 4
P+Q=
4 4
8 5