The document provides information about matrix operations and properties. It defines what a matrix is and different types of matrices. It then discusses operations like addition, subtraction, multiplication of matrices. It also covers properties such as transpose, inverse, adjoint and determinant of a matrix. It provides examples to illustrate matrix operations and properties such as finding the inverse and determinant of given matrices.
3. Matrix: A matrix is a rectangular array of elements or numbers arranged in rows
and columns. Matrix is denoted by first or third parentheses(bracket).
A matrix consists of m horizontal rows and n vertical columns is called 𝑚 × 𝑛
matrix, denoted by
𝐴 =
𝑎11 𝑎12 . . . . . . . 𝑎1𝑛
𝑎21 𝑎22 . . . . . . . 𝑎2𝑛
⋮ ⋮
𝑎𝑚1 𝑎𝑚2 . . . . . . . 𝑎𝑚𝑛
= (𝑎𝑖𝑗)𝑚𝑛
For the entry 𝑎𝑖𝑗, the row number is denoted by 𝑖 and the column number is
denoted by 𝑗. The numbers in a matrix are called its elements.
Example: 𝐴 =
1 5 3
2 4 1
4 3 5
is a 3 × 3 matrix.
4. The size or order of a matrix is described by its number of rows and the
number of columns.
If a matrix 𝑨 has 𝒎 rows and 𝒏 columns then the order of 𝑨 is 𝒎 × 𝒏.
Example: If 𝐴 =
2 1 1
3 4 7
then the order of A is 2 × 3.
5. Row Matrix: A matrix having only a single row is called a row matrix.
Example: 2 3 4
Column Matrix: A matrix having only a single column is called a row matrix.
Example:
5
2
1
Square Matrix: A matrix having equal number of rows and columns is called
a square matrix.
Example:
2 0 −1
5 3 2
1 7 3
Rectangular Matrix: A matrix having equal number of rows and columns is
called a square matrix.
Example:
2 0 −1
1 7 3
Null Matrix: If all elements of a matrix is zero the matrix is called null or zero
matrix and it is shown by 𝟎.
6. Example:
0 0 0
0 0 0
0 0 0
Diagonal Matrix: A square matrix in which all the elements except the main
diagonal are zero is called diagonal matrix.
Example:
2 0 0
0 3 0
0 0 5
Scalar Matrix: In a diagonal matrix if all elements are equal the matrix is
called a scalar matrix.
Example:
3 0 0
0 3 0
0 0 3
Unit/Identity Matrix: A diagonal matrix whose all elements on the main
diagonal are equal to one is called identity or unit matrix. A unit matrix is
usually shown by letter I .
Example: 𝐼 = 𝐼3 =
1 0 0
0 1 0
0 0 1
7. Transpose Matrix: If the rows and columns of a matrix A are interchanged then
the resulting matrix is called transpose of A matrix. It is denoted by 𝐴′/𝐴𝑇.
Example: 𝐴 =
3 2 6
1 5 0
4 3 2
; 𝐴′
= 𝐴𝑇
=
3 2 6
1 5 0
4 3 2
Symmetric Matrix: A matrix A is called symmetric if 𝐴𝑇 = 𝐴.
Example:
1 2
2 4
is a symmetric matrix.
Skew Symmetric Matrix: A matrix A is called skew symmetric if 𝐴 = −𝐴𝑇.
Example:
0 −1
1 0
is a skew symmetric matrix
8. Conditions for Addition and Subtraction of Matrix:
Matrices must have same dimension.
Add/subtract matrices element-by-element
Example: If 𝐴 =
1 3 2
4 2 3
1 5 4
and 𝐵 =
5 2 1
3 1 2
4 3 2
then
𝐴 + 𝐵 =
1 + 5 3 + 2 2 + 1
4 + 3 2 + 1 3 + 2
1 + 4 5 + 3 4 + 2
=
6 5 3
7 3 5
5 8 6
And 𝐴 − 𝐵 =
1 − 5 3 − 2 2 − 1
4 − 3 2 − 1 3 − 2
1 − 4 5 − 3 4 − 2
=
−4 1 1
1 1 1
−3 2 2
Multiplication by scalar: If 𝐴 is a matrix and 𝑘 is any scalar then
𝑘. 𝐴 = 𝑘. (𝑎𝑖𝑗)
𝑚𝑛
This means that all elements of the matrix are multiplied by the scalar 𝑘.
Example: 3
2 3
−1 1
=
6 9
−3 3
9. Multiplication of two matrices 𝑨 and 𝑩, in the form of 𝑨 × 𝑩 or 𝑨𝑩, is
possible if the number of columns in 𝑨 is equal to the number of rows in 𝑩
.
The result of this multiplication is another matrix 𝑪 where the number of
its rows is equal to the number of rows in 𝑨 and number of its columns is
equal to the number of columns in 𝑩; that is:
𝑨𝒎×𝒏 × 𝑩𝒏×𝒑 = 𝑪𝒎×𝒑
For example, if 𝐴 =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
and 𝑋 =
𝑥
𝑦
𝑧
then the multiplication of
A and X will be
AX =
𝑎 𝑏 𝑐
𝑑 𝑒 𝑓
𝑔 ℎ 𝑖
𝑥
𝑦
𝑧
=
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧
𝑑𝑥 + 𝑒𝑦 + 𝑓𝑧
𝑔𝑥 + ℎ𝑦 + 𝑖𝑧
20. The transposed matrix 𝐴 formed by the cofactors of the elements of 𝐴 is
called the adjoint of 𝐴. It is denoted by 𝐴𝑑𝑗 𝐴.
Example: If A=
2 1 0
3 2 1
1 2 0
then the co-factors of 𝐴 are
𝐶11 = +
2 1
2 0
= 0 − 2 = −2; 𝐶12 = −
3 1
1 0
= −(0 − 1) = 1
𝐶13 = +
3 2
1 2
= 6 − 2 = 4; 𝐶21 = −
1 0
2 0
= −(0 − 0) = 0
𝐶22 = +
2 0
1 0
= 0 − 0 = 0; 𝐶23 = −
2 1
1 2
= −(4 − 1) = −3
𝐶31 = +
1 0
2 1
= 1 − 0 = 1; 𝐶32 = −
2 0
3 1
= −(2 − 0) = −2
𝐶33 = +
2 1
3 2
= 4 − 3 = 1
𝐴𝑑𝑗 𝐴 =
−2 1 4
0 0 −3
1 −2 1
𝑇
=
−2 0 1
1 0 −2
4 −3 1
21. Singular and Non-singular Matrix: A matrix 𝐴 is called singular if 𝐴 = 0
and non-singular if |𝐴| ≠ 0.
Example:
1 2
1 2
is singular and
3 2
4 1
is non-singular Matrix.
Inverse Matrix: A matrix 𝐵 is called inverse of a matrix 𝐴 if 𝐴𝐵 = 𝐵𝐴 = 𝐼. It
is denoted by 𝐴−1
.
𝐴−1
=
1
𝐴
𝐴𝑑𝑗 𝐴
Example: Find 𝐴−1 where 𝐴 =
0 1 1
1 2 0
3 −1 4
.
Solution: Here,
𝐴 =
0 1 1
1 2 0
3 −1 4
= 0 − 1 4 − 0 + 1 −1 − 6
= −11 ≠ 0
So, 𝐴 is non-singular. Hence 𝐴 is inversible.
A matrix is inversible only if the matrix is non-singular.
24. The number of linearly independent rows of a matrix is called the rank of a
matrix. It is denoted by 𝝆.
Example: Find the rank of the matrix 𝐴 =
6 2 0 4
−2 −1 3 4
−1 −1 6 10
.
Solution:
6 2 0 4
−2 −1 3 4
−1 −1 6 10
~
−1 −1 6 10
−2 −1 3 4
6 2 0 4
𝑹𝟏 ⟷ 𝑹𝟑
~
1 1 −6 −10
−2 −1 3 4
6 2 0 4
𝑹𝟏
′
= (−𝟏) × 𝑹𝟏
The number of non zero rows in echelon form will be the rank of the
matrix.
Echelon form−
𝟏 𝒂 𝒃
𝟎 𝟏 𝒄
𝟎 𝟎 𝟏
25. ~
1 1 −6 −10
0 1 −9 −16
0 4 −36 −64
𝑹𝟐
′
= −𝟐 × 𝑹𝟏 + 𝑹𝟐
𝑹𝟑
′
= −𝟔 × 𝑹𝟏 − 𝑹𝟑
~
1 1 −6 −10
0 1 −9 −16
0 0 0 0
𝑹𝟑
′
= 𝟒 × 𝑹𝟐 − 𝑹𝟑
The matrix is in echelon form having 2 non-zero rows.
So, Rank of 𝐴, 𝜌 𝐴 = 2.
36. Prepared By
Omar Faruk
Lecturer in Mathematics
Department of Basic Science
World University of Bangladesh
Email: omar.faruk@science.wub.edu.bd
37. Solution of System of Equations by Inverse Matrix
Characteristic Vector and Root
38. Solve the following system of equation by Inverse method
𝒙 − 𝟑𝒚 + 𝟐𝒛 = 𝟑
𝟑𝒙 + 𝟐𝒚 − 𝒛 = 𝟐
𝟐𝒙 − 𝒚 + 𝒛 = 𝟒
Solution: The given system of equation can be written in Matrix-form as
𝟏 −𝟑 𝟐
𝟑 𝟐 −𝟏
𝟐 −𝟏 𝟏
𝒙
𝒚
𝒛
=
𝟑
𝟐
𝟒
Let,
𝐴 =
1 −3 2
3 2 −1
2 −1 1
, B =
3
2
4
and X =
𝑥
𝑦
𝑧
Then the equation reduces to
𝑨𝑿 = 𝑩
43. A non-zero vector 𝑿 is defined as characteristic vector or Eigen vector of a
matrix 𝐴 if there exists a number 𝝀 such that 𝑨𝑿 = 𝝀𝑿 where 𝝀 is defined as
characteristic root or eigen value corresponding to the characteristic vector 𝑿.
The matrix 𝑨 − 𝝀𝑰 is called the characteristic matrix of 𝐴.
The determinant 𝑨 − 𝝀𝑰 is called the characteristic polynomial of 𝐴.
The equation 𝑨 − 𝝀𝑰 = 𝟎 is called the characteristic equation of 𝐴.
Example: If 𝐴 =
1 3 2
4 2 3
1 5 4
then
𝟏 − 𝝀 3 2
4 𝟐 − 𝝀 3
1 5 𝟒 − 𝝀
= 0 is characteristic
equation of 𝐴.
56. Cayley-Hamilton Theorem: Every square matrix satisfies it’s own
characteristic equation.
Explanation: If 𝑨 is an 𝑚 × 𝑛 matrix (where 𝑚 = 𝑛) and 𝑰 is the identity
matrix then the characteristic polynomial of 𝑨 is defined as
If we replace 𝝀 with the matrix 𝑨 then the polynomial will be zero matrix.
Example: Verify Cayley-Hamilton theorem for 𝑨 =
𝟐 𝟐 𝟏
𝟏 𝟑 𝟏
𝟏 𝟐 𝟐
.
Solution: The characteristic polynomial of 𝐴 is,
𝒑 𝝀 = 𝑨 − 𝝀𝑰
𝒑 𝑨 = 𝟎
𝒑 𝝀 = 𝑨 − 𝝀𝑰