11 types of Matrics. For Grade 11 and more

1. Types of Matrices
Narmatha Devi N
Mathematics
Grade 11

2. Here, we will be seeing about some different types of
matrices.
About…

3. • What is a matrix ?
• What is an order of the matrix ?
Recap…

4. • A matrix is a rectangular array or arrangement of entries or
elements displayed in rows and columns put within a square
bracket [ ].
• If a matrix A has m rows and n columns, then it is written as
𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
1 ≤ 𝑖 ≤ 𝑚 , 1 ≤ 𝑗 ≤ 𝑛
Matrix

5. • If a matrix A has m rows and n columns, then the order or size
of the matrix A is defined to be 𝑚 × 𝑛 (read as m by n).
Order of the Matrix
𝐴 =
4 9 7
2 6 1
The Matrix A is of order 2 × 3
Example

6. Type of Matrices…
▪ Row Matrix
▪ Column Matrix
▪ Zero Matrix
▪ Non Zero Matrix
▪ Square Matrix
▪ Diagonal Matrix
▪ Scalar Matrix
▪ Unit Matrix
▪ Upper Triangular Matrix
▪ Lower Triangular Matrix
▪ Triangular Matrix

7. • A matrix having only one row is called a row matrix
• 𝐴 = 𝑎1𝑗 1×𝑛
is a row matrix of order 1 × 𝑛
Row Matrix
𝐴 = 4 7 𝐵 = 2 9 3 𝐶 = 𝑎 𝑏 𝑐 𝑑

8. • A matrix having only one column is called a column matrix
• 𝐴 = 𝑎𝑖1 𝑚×1 is a row matrix of order 𝑚 × 1
Column Matrix
𝐴 =
5
9
𝐵 =
𝑖
𝑗
𝑘
𝐶 =
6
7
3
9

9. • A matrix 𝐴 = 𝑎𝑖𝑗 𝑚×𝑛
is said to be a zero matrix or null
matrix or void matrix denoted by O if
𝑎𝑖𝑗= 0 for all values of 1 ≤ 𝑖 ≤ 𝑚 and 1 ≤ 𝑗 ≤ 𝑛
Zero Matrix / Null Matrix / Void Matrix
𝐴 =
0 0
0 0
𝐵 =
0 0 0
0 0 0
𝐶 =
0
0
0

10. • A matrix A is said to be a non-zero matrix if at least one of the
entries of A is non-zero
Non Zero Matrix
𝐴 = 2 5 𝐵 =
6 0
0 0
𝐶 =
2.5 9.2
0
3
5
8 0

11. • A matrix in which number of rows is equal to the number of
columns, is called a square matrix.
• i.e, a matrix of order 𝑛 × 𝑛 is often referred to as a square
matrix of order n.
Square Matrix
𝐴 =
1 2
3 4
𝐵 =
11 12 13
21 22 23
31 32 33

12. • In a square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
of order n, the elements
𝑎11, 𝑎22, 𝑎33, … 𝑎𝑛𝑛 are called the principal diagonal or
simply the diagonal or main diagonal or leading diagonal
elements.
Square Matrix – the principal diagonal
𝐴 =
5 6.2 3.5
4.8 7 6
12 9 1
,
Here, the principal diagonal elements is 5,7,1

13. • A square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
is said to be a diagonal matrix if
𝑎𝑖𝑗 = 0 whenever i ≠ 𝑗
Diagonal Matrix
𝐴 =
2 0 0
0 5 0
0 0 6
𝐵 =
𝑎 0
0 𝑏
𝐶 =
0 0 0
0 0 0
0 0 0
Note : A square zero matrix is a diagonal Matrix

14. • A diagonal matrix whose entries along the principal diagonal
are equal is called a Scalar matrix.
• A square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
is said to be a Scalar matrix if
𝑎𝑖𝑗 = ቊ
𝑐 𝑖𝑓 𝑖 = 𝑗
0 𝑖𝑓 𝑖 ≠ 𝑗
, where c is a fixed number
Scalar Matrix

15. Scalar Matrix
Note : Any square zero matrix can be considered as a scalar
matrix with scalar 0
𝐴 =
5 0 0
0 5 0
0 0 5
𝐵 =
12 0
0 12
𝐶 =
0 0 0
0 0 0
0 0 0

16. • A square matrix in which all the diagonal entries are 1 and the
rest are all zero is called a unit matrix.
• A square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
is said to be a unit matrix if
𝑎𝑖𝑗 = ቊ
1 𝑖𝑓 𝑖 = 𝑗
0 𝑖𝑓 𝑖 ≠ 𝑗
Unit Matrix
𝐴 =
1 0 0
0 1 0
0 0 1
𝐵 =
1 0
0 1

17. • A square matrix is said to be an upper triangular matrix if all
the elements below the main diagonal are zero.
• A square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
is said to be an upper triangular
matrix if 𝑎𝑖𝑗 = 0 for all 𝑖 > 𝑗
Upper triangular Matrix
𝐴 =
18 50 7
0 1 6
0 0 19
𝐵 =
1 10
0 16
𝐶 =
71 89 0
0 12 4
0 0 35

18. • A square matrix is said to be an lower triangular matrix if all
the elements above the main diagonal are zero.
• A square matrix 𝐴 = 𝑎𝑖𝑗 𝑛×𝑛
is said to be a lower triangular
matrix if 𝑎𝑖𝑗 = 0 for all 𝑖 < 𝑗
Lower triangular Matrix
𝐴 =
91 0 0
45 1.1 0
2.5 5.3 82
𝐵 =
5 0
9 1
𝐶 =
24 0 0
3 8 0
5 7 2

19. • A square matrix which is either upper triangular or lower
triangular is called a triangular matrix
Triangular Matrix
Note : A square matrix that is both upper and lower triangular
simultaneously will turn out to be a diagonal matrix
𝐴 =
5 0 0
7 4 0
8 3 9
𝐵 =
8 9 5
0 4 6
0 0 2
𝐶 =
5 0 0
0 5 0
0 0 5

20. Type of Matrices…
▪ Row Matrix
▪ Column Matrix
▪ Zero Matrix
▪ Non Zero Matrix
▪ Square Matrix
▪ Diagonal Matrix
▪ Scalar Matrix
▪ Unit Matrix
▪ Upper Triangular Matrix
▪ Lower Triangular Matrix
▪ Triangular Matrix

21. Identify the type of each matrix :
𝐴 =
1 2 3
0 4 6
0 0 7
𝐵 =
𝑥
𝑦
𝑧
𝐶 =
1 0 0
0 1 0
0 0 1
𝐷 =
2 0
0 2
Testing time…

22. Thank You…