Chittagong University of Engineering and Technology, Math 233

Lecture Note

Md. Mamun Sarkar

October 2, 2024
 Contents

Contents

1 Matrices and Related Topics 3

1.1 Some imortant Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Is there a matrix which is both row matrix as well as column matrix? . . . . . . 4
1.1.2 Difference between a Matrix and a Determinant . . . . . . . . . . . . . . . . . . 6
1.2 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Elementary Row Operations (transformations) ERO . . . . . . . . . . . . . . . . . . . . 10
1.4 Introduction to Different Elimination Methods . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Gaussian Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Gauss-Jordan Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Reduced Row Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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 1 Matrices and Related Topics

1 Matrices and Related Topics

1.1 Some imortant Definitions

Matrix
A set of mn numbers ( real or complex) arranged in the form of a rectangular array having m
rows and n columns is called a matrix of order 𝑚 × 𝑛 or an 𝑚 × 𝑛 matrix (which is read as m by
n matrix).
An 𝑚 × 𝑛 matrix is usually written as
 𝑎 11 𝑎 12 𝑎 13 · · · 𝑎 1𝑛 
 
 𝑎 21 𝑎 22 𝑎 23 · · · 𝑎 2𝑛 
 
· · · · · · · · · · · · · · · 
 
𝑎𝑚1 𝑎𝑚2 𝑎𝑚3 · · · 𝑎𝑚𝑛 
 

In a compact form the above matrix is represented by (𝑎𝑖 𝑗 ), 𝑖 = 1, 2, 3, · · · , 𝑚; 𝑗 = 1, 2, 3, · · · , 𝑛 or

simply by [𝑎𝑖 𝑗 ]𝑚×𝑛 , where the symbols 𝑎𝑖 𝑗 represents any numbers (The element/entry 𝑎𝑖 𝑗 lies
in the ith row (from top) and jth column (from left)).
Notations A matrix is usually denoted by capital letters such as A, B, C, etc.
Note:
1. A matrix may also be represented by the symbols [𝑎𝑖 𝑗 ], (𝑎𝑖 𝑗 ), ||𝑎𝑖 𝑗 ||.
2. The numbers 𝑎 11, 𝑎 12, · · · etc., of rectangular array are called the elements or entries of
the matrix.
3. A matrix is essentially an arrangement of elements and has no value.
4. The plural of ’Matrix’ is ’Matrices’.

Row Matrix or Row vector

A matrix 𝐴 is said to be a row matrix or a row vector if it contains only one row, i.e. a matrix
𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is said to be a row matrix if 𝑚 = 1.

Column Matrix or Column vector

A matrix 𝐴 is said to be a column matrix or a column vector if it contains only one column, that
is, a matrix 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is said to be a column matrix if 𝑛 = 1.

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1.1.1 Is there a matrix which is both row matrix as well as column matrix?
Square Matrix

A matrix 𝐴 is said to be square if the number of rows and the number of columns are equal, that
is, 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is called a square matrix if 𝑚 = 𝑛.

Remark. If 𝐴 is a square matrix of order n, the elements (entries) 𝑎 11, 𝑎 22, 𝑎 33, · · · , 𝑎𝑛𝑛 are said to
construct the diagonal elements of matrix A. The line along which the diagonal elements lie is called
the principal diagonal or leading diagonal.

Diagonal Matrix

A matrix 𝐴 is said to be ’Diagonal’ if all its non-diagonal elements are zero, that is, 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛
is called a diagonal matrix if 𝑎𝑖 𝑗 = 0 when 𝑖 ≠ 𝑗.

3 0 0
−1 0
 
𝐴 = 2 ;𝐵 = ; 𝐶 = 0 5 0 are diagonal matrix of order 1,2 and 3 respectively.
   
0 2 0 0 7
A diagonal matrix of order n having 𝑑 1, 𝑑 2, · · · , 𝑑𝑛 as diagonal elements may be denoted by
 

𝑑𝑖𝑎𝑔(𝑑 1, 𝑑 2, · · · , 𝑑𝑛 ). Thus 𝐴 = 𝑑𝑖𝑎𝑔(2), 𝐵 = 𝑑𝑖𝑎𝑔(−1, 2), 𝐶 = 𝑑𝑖𝑎𝑔(3, 5, 7).

1. No element of principal diagonal in a diagonal matrix is zero.
2. The minimum number of zeros in a diagonal matrix is given by 𝑛(𝑛 − 1), where 𝑛 is the
order of the matrix.
3. 𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧) + 𝑑𝑖𝑎(𝑝, 𝑞, 𝑟 ) = 𝑑𝑖𝑎𝑔(𝑥 + 𝑝, 𝑦 + 𝑞, 𝑧 + 𝑟 )
4. 𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧) × 𝑑𝑖𝑎𝑔(𝑝, 𝑞, 𝑟 ) = 𝑑𝑖𝑎𝑔(𝑥𝑝, 𝑦𝑞, 𝑧𝑟 )
5. (𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧) −1 ) = 𝑑𝑖𝑎𝑔(1/𝑥, 1/𝑦, 1/𝑧)
6. (𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧))𝑡 = 𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧)
7. (𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧))𝑛 = 𝑑𝑖𝑎𝑔(𝑥 𝑛 , 𝑦𝑛 , 𝑧𝑛 )
8. Eigen value of 𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧) = 𝑥, 𝑦& 𝑧
9. Determinant of 𝑑𝑖𝑎𝑔(𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧

Scalar Matrix
A diagonal matrix is said to be a scalar matrix,( if its diagonal elements are equal. Thus, 𝐴 =
0 if 𝑖 ≠ 𝑗
(𝑎𝑖 𝑗 )𝑚×𝑛 is called a scalar matrix, if 𝑎𝑖 𝑗 =
𝑘 if 𝑖 = 𝑗

Unity or Identity Matrix

A diagonal matrix is said to be an identity matrix, if its(diagonal elements are equal to 1. Thus,
0, if 𝑖 ≠ 𝑗
𝐴 = (𝑎𝑖 𝑗 )𝑛×𝑛 is called a unit or identity matrix, if 𝑎𝑖 𝑗 = .
1, if 𝑖 = 𝑗

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Unity or Identity Matrix

1 0
 
A unit matrix of order n is denoted by 𝐼𝑛 or by 𝐼 . For example, 𝐼 1 = [1], 𝐼 2 = ,
0 1
1 0 0
𝐼 3 = 0 1 0 are identity matrices of order 1,2 and 3, respectively.
 
0 0 1
 

Singleton Matrix

A matrix is said to be a singleton matrix if it has only one element, i.e. a matrix 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is
said to be a singleton matrix if, if 𝑚 = 𝑛 = 1.

For example, [3], [𝑘], [−2] are singleton matrices.

Determinant of square matrix

Let 𝐴 = (𝑎𝑖 𝑗 )𝑛×𝑛 be a matrix. The determinant formed by the elements of A is said to be the
determinant of the matrix 𝐴. This is denoted by |𝐴|.

Properties:
• If 𝐴1, 𝐴2, · · · , 𝐴𝑛 are square matrices of the same order, then

|𝐴1, 𝐴2, · · · , 𝐴𝑛 | = |𝐴1 |, |𝐴2 |, · · · , |𝐴𝑛 |

.
• If k is a scalar and 𝐴 is a square matrix of order n, then |𝑘𝐴| = 𝑘 𝑛 |𝐴|.
• |𝐴 0 | = |𝐴|
• |𝐴𝐵| = |𝐴||𝐵| and |𝐴𝐵| = |𝐵𝐴|
• If 𝐴 is orthogonal matrix, then |𝐴| = ±1
• If 𝐴 is a skew-symmetric matrix of odd order, then |𝐴| = 0.
• If 𝐴 is a skew-symmetric matrix of even order, then |𝐴| is a perfect square.
• |𝐴𝑛 | = |𝐴|𝑛 , where 𝑛 ∈ 𝑁
• If 𝐴 = 𝑑𝑖𝑎𝑔(𝑎 1, 𝑎 2, 𝑎 3, · · · , 𝑎𝑛 ), then |𝐴| = 𝑎 1, 𝑎 2, 𝑎 3, · · · , 𝑎𝑛

Comparable Matrices

Two matrices 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 and 𝐵 = (𝑏𝑖 𝑗 )𝑝×𝑞 are said to be comparable, if 𝑚 = 𝑝 and 𝑛 = 𝑞.
   
𝑎 𝑏 𝑐 𝑝 𝑞 𝑟
𝐴= ,𝐵 =
𝑑 𝑒 𝑓 𝑠 𝑡 𝑢

Transpose of a Matrix

Let 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 be any given matrix, then the matrix obtained by interchanging the rows and
columns of 𝐴 is called the transpose of 𝐴. Transpose of matrix 𝐴 is denoted by 𝐴 0 or 𝐴𝑇 or 𝐴𝑡 .
In other words, if 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 , then 𝐴 0 = (𝑎𝑖 𝑗 )𝑚×𝑛 .

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1.1.2 Difference between a Matrix and a Determinant

1. A matrix cannot be reduced to a number but determinant can be reduced to a number.
2. The number of rows may or may not be equal to the number of columns in matrices but in
determinant the number of rows is equal to the number of columns.
3. On interchanging the rows and columns, a different matrix is formed but in determinant it does
not change the value.
4. A square matrix 𝐴 such that |𝐴| ≠ 0, is called a non-singular matrix. If |𝐴| = 0, then the matrix is
called a singular matrix.
5. Matrices are represented by [ ], ( ), || ||, but the determinant is represented by | |.
Remark. If two matrices 𝐴 and 𝐵 are of the same order, then only their addition and subtraction is possible
and these matrices are said to be conformable for addition or subtraction. On the other hand, if the matrices
𝐴 and 𝐵 are of different orders, then their addition and subtraction are not possible and these matrices are
called non-conformable for addition or subtraction.
Conformable for Multiplication: Let 𝐴 and 𝐵 be two matrices that are said to be conformable to the
product AB if the number of columns in 𝐴 (called the pre-factor) is equal to the number of rows in 𝐵 (called
the post-factor). Otherwise, they are non-conformable for multiplication. Thus,
1. 𝐴𝐵 is defined, if the number of columns in 𝐴 = number of rows in 𝐵.
2. 𝐵𝐴 is defined, if the number of columns in 𝐵 = number of rows in 𝐴.

1.2 Special Matrices

Idempotent Matrix

A square matrix 𝐴 is called an idempotent matrix, provided it satisfies the relation 𝐴2 = 𝐴.

Note: 𝐴𝑛 = 𝐴 ∀𝑛 ≥ 2, 𝑛 ∈ 𝑁 .
2 −2 −4
Example, 𝐴 = −1 3 4 

1 −2 −3


Periodic Matrix

A square matrix 𝐴 is called periodic, if 𝐴𝑘+1 = 𝐴, where 𝑘 is the least positive integer. If 𝐾 is the
least positive integer for which 𝐴𝑘+1 = 𝐴, then 𝑘 is said to be the period of a 𝐴.

For 𝑘 = 1, we get 𝐴2 = 𝐴 and we called it to be idempotent matrix.

Note: period of a matrix is 1.

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Nilpotent Matrix

A square matrix 𝐴 is called nilpotent matrix of order m provided it satisfies the relation 𝐴𝑘 = 0
and 𝐴𝑘−1 ≠ 0, where 𝑘 is positive integer and 𝑂 is null matrix and 𝑘 is the order of the nilpotent
matrix.

1 1 3 
5 2 6 

𝐴=

−2 −1 −3
 

Involutory Matrix

A square matrix 𝐴 is called involutory matrix provided it satisfies the relation 𝐴2 = 𝐼 , where 𝐼 is
identity matrix.

−5 −8 0 
𝐴 =  3 5 0 
 
1 2 −1

Note: 𝐴 = 𝐴−1 for an involutory matrix.

Symmetric Matrix

A square matrix 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is said to be symmetric if 𝐴𝑇 = 𝐴 i.e. 𝑎𝑖 𝑗 = 𝑎 𝑗𝑖 ∀𝑖, 𝑗

𝑎 ℎ 𝑔  𝑎 ℎ 𝑔 
For example, if 𝐴 = ℎ 𝑏 𝑓  then 𝐴 = ℎ 𝑏 𝑓  ’Ṫhus we can say that A is a symmetric
  𝑇
 
 
𝑔 𝑓 𝑐  𝑔 𝑓 𝑐 
   
matrix as 𝐴 = 𝐴.
𝑇

Note:
𝑛(𝑛 − 1)
i. Maximum number of distinct entries in any symmetric matrix of order n is
2
ii. for any square matrix 𝐴 with real number entries, then 𝐴 + 𝐴 0 is a symmetric matrix.

Skew-symmetric Matrix

A square matrix 𝐴 = (𝑎𝑖 𝑗 )𝑚×𝑛 is said to be skew-symmetric if 𝐴𝑇 = −𝐴, i.e. 𝑎𝑖 𝑗 = −𝑎 𝑗𝑖 ∀𝑖, 𝑗

0 ℎ 𝑔  0 −ℎ −𝑔  0 ℎ 𝑔 
For example, if 𝐴 = −ℎ 0 𝑓  then 𝐴 = ℎ 0 −𝑓  = − −ℎ 0 𝑓  = −𝐴. Thus we
 𝑇
  
−𝑔 −𝑓 0  𝑔 𝑓 0  −𝑔 −𝑓 0 
    
can say that A is a skew-symmetric matrix as 𝐴 = −𝐴.
𝑇

Note:
i. Trace of a skew-symmetric matrix is always zero.
ii. For any square matrix 𝐴 with real number entries, then 𝐴 − 𝐴 0 is a skew-symmetric matrix.
iii. Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix. i.e., if 𝐴 is a square matrix, then we can write,
1 1
𝐴= (𝐴 + 𝐴 0) + (𝐴 − 𝐴 0)
2 2
.

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Orthogonal Matrix

A square matrix 𝐴 is said to be an orthogonal matrix, iff 𝐴𝐴 0 = 𝐼 , where 𝐼 is an identity matrix.

Note:
1. If 𝐴𝐴 0 = 𝐼 , then 𝐴−1 = 𝐴 0.
2. The sum of squares of the elements of a row is unity.
3. The sum of products of corresponding elements of different row is zero.
4. |𝐴| = ±1 (If |𝐴| = 1 then 𝐴 is said to be proper orthogonal Matrix otherwise it is called
improper orthogonal Matrix)
5. If 𝐴 and 𝐵 are orthogonal, then 𝐴𝐵 is also orthogonal.
6. If 𝐴 is orthogonal, then 𝐴−1 and 𝐴 0 are also orthogonal.

 0 2𝛽 𝛾 
1. If 𝛼 𝛽 −𝛾  is orthogonal, then find the value of 2𝛼 2 + 6𝛽 2 + 3𝛾 2 .
 
𝛼 −𝛽 𝛾 
 
1 2 2 
2. If 𝐴 = 2 1 −2 is a matrix satisfying 𝐴𝐴 0 = 9𝐼 3 , find the value of |𝑎| + |𝑏 |.
 
𝑎 2 𝑏 
 
0 2𝑚 𝑛 
3. Find 𝑙, 𝑚, 𝑛 and if 𝐴 =  𝑙 𝑚 −𝑛  is orthogonal
 
𝐴−1
 𝑙 −𝑚 𝑛 
 
Complex conjugate (or conjugate) of a Matrix

If a matrix 𝐴 has complex numbers as its elements, the matrix

√ obtained from 𝐴 by replacing each
element of 𝐴 by its conjugate (𝑎 ± 𝑖𝑏 = 𝑎 ∓ 𝑖𝑏 where𝑖 = −1) is called the conjugate of matrix 𝐴
and is denoted by 𝐴.¯

2 + 5𝑖 3 − 𝑖 7  2 − 5𝑖 3 + 𝑖 7 
For example, if 𝐴 =  −2𝑖 6 + 𝑖 7 − 5𝑖  then 𝐴¯ =  2𝑖 6 − 𝑖 7 + 5𝑖 
 

1−𝑖 3 6𝑖   1+𝑖 3

  −6𝑖 

Conjugate transpose Matrix

The conjugate of the transpose of a matrix 𝐴 is called the conjugate transpose of 𝐴 and is denoted
by 𝐴𝜃 , i.e., 𝐴𝜃 = conjugate of 𝐴𝑇 = 𝐴𝑇 .

Hermitian Matrix

A square matrix is said to be hermitian if 𝐴𝜃 = 𝐴, i.e., 𝑎𝑖 𝑗 = 𝑎 𝑗𝑖 ∀𝑖, 𝑗.

 𝛼 𝜆 + 𝑖𝜇 𝜃 + 𝑖𝜙 
Example: 𝐴 = 𝜆 − 𝑖𝜇

𝛽 𝑥 + 𝑖𝑦 
𝜃 − 𝑖𝜙 𝑥 − 𝑖𝑦 𝜆 


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Skew-Hermitian Matrix

A square matrix is said to be skew-hermitian if 𝐴𝜃 = −𝐴, that is, 𝑎𝑖 𝑗 = −𝑎 𝑗𝑖 ∀𝑖, 𝑗.

 2𝑖 −2 − 3𝑖 −2 + 𝑖 
𝐴 = 2 − 3𝑖 3𝑖  .

 −𝑖
 2+𝑖 3𝑖 0 
Note: All diagonal elements of a skew-hermitian matrix are either zeros or purely imaginary.


Unitary MAtrix

A square matrix 𝐴 is said to be a unitary matrix iff 𝐴𝐴𝜃 = 𝐼 , where 𝐼 is an identity matrix.

Note:
1. If 𝐴𝐴𝜃 = 𝐼 ,then 𝐴−1 = 𝐴𝜃
2. If 𝐴 and 𝐵 are unitary, then 𝐴𝐵 is also unitary.
3. If 𝐴 is unitary, then 𝐴−1 and 𝐴 0 are also unitary.

Adjoint of a Matrix

Let 𝐴 = 𝑎𝑖 𝑗 be a square matrix of order n and let 𝐶𝑖 𝑗 be the cofactor matrix of 𝑎𝑖 𝑗 in 𝐴. Then, the
transpose of cofactors of elements of 𝐴 is called the adjoint of 𝐴 and is denoted by adj(𝐴).

Thus, adj(𝐴) = [𝐶𝑖 𝑗 ] 0 =⇒ (𝑎𝑑 𝑗 𝐴)𝑖 𝑗 = 𝐶 𝑗𝑖 = cofactor of 𝑎𝑖 𝑗 in 𝐴.

Problems
−1 1 1 
1. If 𝐴 =  1 −1 1  , find the values of

1 1 −1


i. |𝐴||adj 𝐴| ii. |adj(3𝐴)|

ii. |adj(adj(adj 𝐴))| iv. adj(adj 𝐴)
3 −3 4
2. If 𝐴 = 2 −3 4 and 𝐵 is the adjoint of 𝐴, find the value of |𝐴𝐵 + 2𝐼 |, where 𝐼 is the
 
0 −1 1
identity matrix of order 3.
 

Book: Skills in Mathematics- Example 32, 33; Page-628.

Shortcut for Adjoint in page no. 627.

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Inverse of a Matrix (reciprocal Matrix)

A square matrix 𝐴 (non-singular) of order n is said to be invertible if there exists a square matrix
𝐵 of the same order such that 𝐴𝐵 = 𝐼𝑛 = 𝐵𝐴, then 𝐵 is called the inverse (reciprocal) of 𝐴 and is
denoted by 𝐴−1 .
𝑎𝑑 𝑗 𝐴
∴ 𝐴−1 = , provided|𝐴| ≠ 0
|𝐴|

• (Uniqueness of inverse) Inverse of a matrix if exists is unique.

• The matrix under consideration should be a square matrix.
• The necessary and sufficient condition for a square matrix 𝐴 to be invertible is |𝐴| ≠ 0.

Upper and Lower triangular Matrix

A square matrix in which all elements below the principal diagonal are zero is called an upper
triangular matrix i.e., a square matrix 𝐴 is said to be an upper triangular matrix if 𝐴 = (𝑎𝑖 𝑗 )𝑛 , if
𝑎𝑖 𝑗 = 0, ∀𝑖 > 𝑗. On the other hand, a square matrix in which all elements above the principal
diagonal are zero is called the lower triangular matrix, that is, 𝐴 is called a lower triangular
matrix. if 𝑎𝑖 𝑗 = 0, ∀𝑖 < 𝑗,

1 5 9  1 0 0
For example, 𝐴 = 0 7 −1 is an upper triangular matrix and 𝐵 = 5 7 0 is a lower
   
 
0 0 5  9 −1 5
triangular matrix.
   

Note:
𝑛(𝑛 − 1)
Minimum number of zeroes in a triangular matrix is given by , where 𝑛 is the order of
2
the matrix.

1.3 Elementary Row Operations (transformations) ERO

The following three types of operations (transformations) on the rows of a given matrix are known as
elementary row operations.
i. Interchange of any two rows- The interchange of 𝑖th and 𝑗th row is denoted by 𝑅𝑖 ↔ 𝑅 𝑗 or 𝑅𝑖 𝑗 .
ii. Multiplication of the elements of a row by a nonzero number - Suppose that each element of the
𝑖th row of a given matrix is multiplied by a nonzero number 𝑘. Then we denote it by 𝑅𝑖 → 𝑘𝑅𝑖 or
𝑅𝑖 (𝑘).
iii. Multiplying each element of a row by a nonzero number and then adding them to the correspond-
ing elements of another row- Suppose each element of 𝐽 th row of a matrix 𝐴 is multiplied by a
nonzero number 𝑘 and then added to the corresponding elements of 𝑖th row. We denote it by
𝑅𝑖 → 𝑅𝑖 + 𝑘𝑅 𝑗 or 𝑅𝑖 𝑗 (𝑘).

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Equivalent Matrices

Two matrices are said to be equivalent if one is obtained from the other by elementary operations
(transformations). Thus, if a matrix 𝐵 is obtained from a matrix 𝐴 by one or more elementary
operations, then 𝐵 is said to be equivalent to 𝐴. The symbol ∼ is used for the equivalence, i.e.
𝐴 ∼ 𝐵.

(i) If 𝐴 and 𝐵 are equivalent matrices, there exist non-singular matrices 𝑃 and 𝑄 such that
𝐵 = 𝑃𝐴𝑄.
(ii) If 𝐴 and 𝐵 are equivalent matrices such that 𝐵 = 𝑃𝐴𝑄, then 𝑃 −1 𝐵𝑄 −1 = 𝐴.
(iii) Every non-singular square matrix can be expressed as the product of elementary matrices.

1.4 Introduction to Different Elimination Methods

1.4.1 Gaussian Elimination Method

If 𝐴 be a non-singular matrix of order n, then write 𝐴 = 𝐼𝑛 𝐴. If 𝐴 is reduced to 𝐼𝑛 by elementary
operations (LHS), then suppose 𝐼𝑛 is reduced to P(RHS) and not change A in RHS, then after elementary
operations, we get
𝐼𝑛 = 𝑃𝐴
then P is the inverse of 𝐴,
𝑃 = 𝐴−1

The most common problem encountered in practice is the one in which there are n equations as well as
n unknowns-called a Square System-for which there is a unique solution. Since Gaussian elimination
is a straightforward for this case, we begin by detailed description of Gaussian elimination as applied to
the following simple (but typical) square system:

2𝑥 + 𝑦 + 𝑧 = 1
6𝑥 + 2𝑦 + 𝑧 = −1 (1.1)
−2𝑥 + 2𝑦 + 𝑧 = 7
In every step, we need to find the pivot position and eliminate all the terms below this position using
the ERO. The coefficient in the pivot position is called a pivotal element (or simply a pivot), While
the equation containing the pivot element is called the Pivotal equation. Only nonzero numbers
are allowed to be pivots. If a coefficient in a pivot position is over 0, then the pivotal equation is
interchanged with an equation below the pivotal equation to produce a nonzero pivot. (This is always
possible for square systems possessing a unique solution.) Unless it is 0, the first coefficient of the first
coefficient of the first equation is taken as the first pivot. For example, the circled 2 in the system
below is the pivot for the first step:
2 𝑥 +𝑦 +𝑧 = 1
6𝑥 + 2𝑦 + 𝑧 = −1 (1.2)
−2𝑥 + 2𝑦 + 𝑧 = 7
Step-1: Eliminate all the terms below the first pivot Subtract three times the first equation from the
second and add the first equation to the third equation to produce the equivalent system:(𝐸 2 −
3𝐸 1 )&(𝐸 3 + 𝐸 1 )
2 𝑥 +𝑦 +𝑧 = 1
−𝑦 − 2𝑧 = −4 (1.3)
3𝑦 + 2𝑧 = 8

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Step-2: Select a new pivot. Select a new pivot by moving down and to the right1 . If this coefficient is
not 0, then it is the next pivot. Otherwise, interchange with an equation below this position so as
to bring a nonzero number into this pivotal position. In our example, −1 is the second pivot as
identified below:
𝑥 +𝑦 +𝑧 = 1
-1 𝑦 − 2𝑧 = −4 (1.4)
3𝑦 + 2𝑧 = 8

Step-3: Eliminate all terms below the second pivot Add three times the second equation to the third
equation so as to produce the equivalent system: (𝐸 3 + 3𝐸 2 )
𝑥 +𝑦 +𝑧 = 1
-1 𝑦 − 2𝑧 = −4 (1.5)
−4𝑧 = −4

• In general, at each step you move down and to the right to select the next pivot, then eliminate
all terms below the pivot until no longer can be proceed. In this example, the third pivot is −4,
but since there is nothing below the third pivot to eliminate, the process is complete.
At this point, the system is said to be triangularized. A triangular system can be easily solved by
back substitution. For example, solve the last equation in 1.5 to obtain
𝑧=1

Substitute 𝑧 = 1 back into the second equation in equation 1.5 and determine
𝑦 = 4 − 2𝑧 = 4 = 2(1) = 2.
Finally, substitute 𝑧 = 1 and 𝑦 = 2 back into the first equation in equation 1.5 to get
1 1
𝑥= (1 − 𝑦 − 𝑧) = (1 − 2 − 1) = −1,
2 2
which completes the solution.
Now if we write by discarding the symbols ”𝑥”, ”𝑦”, ”𝑧” and ” = ” then a system of linear equations
reduced to a rectangular array of numbers where each horizontal line represents one equation. Thus
2 1 1 1
the system in 1.1 reduces to: ­ 6 2 1 −1 ® (The line emphasizes where = appeared.)
© ª

« −2 2 1 7 ¬
The array of coefficients- the numbers on the left-hand side of the vertical line-is called the coefficient
matrix for the system. The entire array- the coefficient matrix augmented by the numbers from the
right-hand side of the system- is called the augmented matrix associated with the system. If the
coefficient matrix be denoted by A and the right-hand side is denoted by b, then the augmented matrix
associated with the system is denoted by [A|b]. Note: Gaussian elimination can be executed on the
associated augmented matrix [A|b] by performing an elementary operation on the rows of [A|b].

Submatrix
A submatrix of a given matrix 𝐴 is an array obtained by deleting any combination of rows and
columns from 𝐴.

1. Solve the following systems using Gaussian elimination with back substitution:
1 Thestrategy of selecting a pivots in numerical computation is usually bit more complicated than simply using the next
coefficient that is down and to the right

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𝑣 −𝑤 = 3 4𝑥 2 − 3𝑥 3 = 3
i. −2𝑢 + 4𝑣 − 𝑤 = 1 iv. −𝑥 1 + 7𝑥 2 − 5𝑥 3 = 4
−2𝑢 + 5𝑣 − 4𝑤 = −2 −𝑥 1 + 8𝑥 2 − 6𝑥 3 = 5
𝑣 −𝑤 = 3 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 1
ii. −2𝑢 + 4𝑣 − 𝑤 = 1 𝑥 1 + 𝑥 2 + 3𝑥 3 + 3𝑥 4 = 3
v.
−2𝑢 + 5𝑣 − 4𝑤 = −2 𝑥 1 + 𝑥 2 + 2𝑥 3 + 3𝑥 4 = 3
2𝑥 1 − 𝑥 2 = 0 𝑥 1 + 3𝑥 2 + 3𝑥 3 + 3𝑥 4 = 4
iii. −𝑥 1 + 2𝑥 2 − 𝑥 3 = 0
−𝑥 2 + 𝑥 3 = 1

2.
vi. Consider the following three systems where the coefficients are the same foe each system, but
4𝑥 − 8𝑦 + 5𝑧 = 1 0 0
the right-hand sides are different (this situation occurs frequently): 4𝑥 − 7𝑦 + 4𝑧 = 0 −1 0
3𝑥 − 4𝑦 + 2𝑧 = 0 0 7
Solve all three systems at one time by performing Gaussian elimination on an augmented matrix
of the form [ 𝐴 𝑏 1 𝑏 2 𝑏 3 ]
3. Attempt to solve the system
−𝑥 1 + 3𝑥 2 − 2𝑥 3 = 4
−𝑥 1 + 4𝑥 2 − 3𝑥 3 = 5
−𝑥 1 + 5𝑥 2 − 4𝑥 3 = 6
using Gaussian elimination and explain why this system must have infinitely many solutions.
4. Explain why a linear system can never have exactly two different solutions. Explain your argument
to explain the fact that if a system has more than one solution, then it must have infinitely many
different solutions.

1.4.2 Gauss-Jordan Method

This method is a variation of the well known Gaussian elimination method with two distinguish features:
• At each step the pivot element is forced to be 1.
• At each step all terms above the pivot as well as all terms below the pivot are eliminated and thus
circumvents the use of back substitution.
1. Apply the Gauss-Jordan method to solve the following system:

2𝑥 1 + 2𝑥 2 + 6𝑥 3 = 4
2𝑥 1 + 𝑥 2 + 7𝑥 3 = 6
−2𝑥 1 − 6𝑥 2 − 7𝑥 3 = −1

Solution.
2 2 6 4 1 1 3 2
𝑅1
­ 2 1 7 6 ® ∼ ­ 2 1 7 6 ® 𝑅2 −2𝑅1
© ª © ª
𝑅3 +2𝑅1
2
« −2 −6 −7 −1 ¬ « −2 −6 −7 −1 ¬
1 1 3 2 1 1 3 2
∼ ­ 0 -1 1 2 ® ∼ ­ 0 1 −1 −2 ® 𝑅1 −𝑅2
© ª © ª
(−𝑅2 ) 𝑅3 +4𝑅2
« 0 −4 −1 3 ¬ « 0 −4 −1 3 ¬

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1 0 4 4 1 0 4 4
∼ ­ 0 1 −1 −2 ® ∼ ­ 0 1 −1 −2 ® 𝑅1 −4𝑅3
© ª © ª
−𝑅3 /5 𝑅2 +𝑅3
« 0 0 -5 −5 ¬ « 0 0 1 1 ¬
1 0 0 0
∼ ­ 0 1 0 −1 ®
© ª

« 0 0 1 1 ¬
Therefore the solution is
𝑥 1   0 
   
𝑥 2  = −1
𝑥 3   1 
   
   

Remark. Gauss-Jordan requires more arithmetic than Gaussian elimination with back substitution-
approximately 50%. For small systems, this does not show a great deal of difference, but in practical work
the Gauss-Jordan method is not recommended for solving linear systems. It does have some theoretical
advantages, especially in matrix inversion.
1. Use the Gauss-Jordan method to solve the following systems:

4𝑥 2 − 3𝑥 3 = 3 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 1
i. −𝑥 1 + 7𝑥 2 − 5𝑥 3 = 4 𝑥 1 + 2𝑥 2 + 2𝑥 3 + 2𝑥 4 = 0
ii.
−𝑥 1 + 8𝑥 2 − 6𝑥 3 = 5 𝑥 1 + 2𝑥 2 + 3𝑥 3 + 3𝑥 4 = 0
𝑥 1 + 2𝑥 2 + 3𝑥 3 + 4𝑥 4 = 0

2. Use the Gauss-Jordan method to solve the following three systems at the same time.

2𝑥 1 − 𝑥 2 = 1 0 0
−𝑥 1 + 2𝑥 2 − 𝑥 3 = 0 1 0
− 𝑥2 + 𝑥3 = 0 0 1

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Row Echelon Form (REF)

An 𝑚 × 𝑛 matrix with rows 𝐸𝑖∗ and columns 𝐸 ∗𝑗 is said to be in row echelon form provided that
the following two conditions are met.
• If 𝐸𝑖∗ consists entirely of zeros, then all rows below 𝐸𝑖∗ are also entirely zero; i.e., all zero
rows are at the bottom.
• If the first non-zero entry in 𝐸𝑖∗ lies in position 𝑗 th , then all entries below the position 𝑖 th
in columns 𝐸 ∗1, 𝐸 ∗2, · · · , 𝐸 ∗𝑗 are zero.
These two conditions say that the nonzero entries in an echelon form must lie on or above a
stair-step line that emanates from the upper left-hand corner and slopes down and to the right.
The pivots are the first nonzero entries in each row. A typical structure for a matrix in row
echelon from is illustrated below with the pivots circled.

* ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 *

∗ ∗ ∗ ∗ ∗
0 0 0

* ∗ ∗ ∗ ∗
0 0 0 0 0 0

 * ∗
0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

The positions of the pivots in E (and A) are uniquely determined by the entries in A. Because of
the pivotal positions are unique, it follows that the number of pivots, which is the same as the
number of nonzero rows in E, is also uniquely determined by the entries in A. This number is
called the rank of A.

Rank of a Matrix
Suppose 𝐴𝑚×𝑛 is reduced by row operations to echelon form 𝐸. The rank of A is defined as the
number
𝑟𝑎𝑛𝑘 (𝐴) = number of pivots
= number of nonzero rows in 𝐸
= number of basic columns in 𝐴
where the basic columns of A are defined to be those columns in A that contain the pivotal
positions.

It is denoted by 𝑟𝑘 (𝐴) or by 𝜌 (𝐴).

Determination of Rank of a Matrix:
I. Enumeration: Evaluate all minors such that a minor of order 𝑟 is non-zero and every
minor of (𝑟 + 1) or more is zero.
Note: This is impracticable for matrices of higher order.
II. Apply only elementary row operations on 𝐴. Then the number of non-zero rows is the
rank of A.
III. Using Normal form.
IV. Echelon Form: the number of non-zero rows in an echelon form is the rank.
Remark. Equivalent matrices have the same order and rank because elementary transformations
do not alter (effect) their order and rank.

1.5 Reduced Row Echelon Form

At each step of the Gauss-Jordan method, the pivot is forced to be a 1, and then all entries above and
below the pivotal 1 are annihilated. If A is the coefficient matrix for a square system with a unique

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solution, then the end result of applying the Gauss-Jordan method to A is a matrix with 1’s on the main
diagonal and 0’s everywhere else. That is,
1 0 · · · 0
text above 0 1 · · · 0
 
 
𝐴 −−−−−−−→  .. .. . . .
. . . .. 
0 0 · · · 1

 
But if the Gauss-Jordan technique is applied to a more general 𝑚 × 𝑛 matrix then the final result is not
necessarily the same as described above.
Reduced Row Echelon Form
A matrix 𝐸𝑚×𝑛 is said to be in reduced row echelon form provided that the following three
conditions hold.
• E is in row echelon form
• The first nonzero entry in each row (i.e., each pivot) is 1.
• All entries above each pivot are 0
A typical structure for a matrix in reduced row echelon form is illustrated below, where entries
marked * can be either zero or nonzero numbers:
 1 ∗ 0 0 ∗ ∗ 0 ∗
0 0 1 0 0

 ∗ ∗ ∗
0 0 0 1 0

 ∗ ∗ ∗
0 0 0 0 0 0


 1 ∗

 0 0 0 0 0 0 0 0

 0 0 0 0 0 0 0 0

For a matrix 𝐴, the symbol 𝐸𝐴 will hereafter denote the unique reduced row echelon form derived
from 𝐴 by means of row operations.

This is also called the Hermite normal form. The only difference between the RREF and NF is
that, ERO is applied in case of RREF whereas ERO and ECO both are applied in case of NF.

1. Determine 𝐸𝐴 , the 𝑟𝑎𝑛𝑘 (𝐴) and identify the basic columns of

1 2 2 3 1
­2 4 4 6 2®
© ª
𝐴=­
­3 6 6 9 6®
®

«1 2 4 5 3¬

Solution.
1 2 2 3 1 1 2 2 3 1 1 2 2 3 1
­2 4 4 6 2® ­ 0 0 0 0 0® ­ 0 0 2 2 2®
© ª © ª © ª
®∼­ ®∼­
­3 6 6 9 6® ­ 0 0 0 0 3® ­ 0 0 0 0 3®
­ ®

«1 2 4 5 3¬ « 0 0 2 2 2¬ « 0 0 0 0 0¬
1 2 2 3 1 1 2 0 1 −1 1 2 0 1 −1
­0 0 1 1 1® ­ 0 0 1 1 1® ­0 0 1 1 1®
© ª © ª © ª
∼­ ®∼­ ®∼­
­0 0 0 0 3® ­ 0 0 0 0 3® ­0 0 0 0 1®
®

«0 0 0 0 0¬ « 0 0 0 0 0¬ «0 0 0 0 0¬
1 2 0 1 0
­0 0 1 1 0®
© ª
∼­
­0 0 0 0 1®
®

«0 0 0 0 0¬

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Therefore, 𝑟𝑎𝑛𝑘 (𝐴) = 3 and {𝐴∗1, 𝐴∗3, 𝐴∗5 } are the three basic columns.
Remark. [A summary]
• A matrix has an infinite number of EF
• A matrix has only one RREF
• A matrix has only one NF
• Let 𝐴 be any matrix. To find an EF of 𝐴 use ERO only and:
– select 1𝑠𝑡 pivot
– make all entries zero below with the help of this pivot
– select 2𝑛𝑑 pivot
– make all entries zero with the help of this pivot
– Continue this process until an EF appears.
• To find the RREF of 𝐴 use only ERO and:
– Starting with an previously obtained EF make all entries above the 2𝑛𝑑 pivot zero with the help
of this pivot.
– follow similar steps for other pivots as well.
– make each pivot 1.
• To find the NF, use ERO or ECO or both
– Make all the entries right to the pivots zero with the help of that pivot using ECO.
– Repeat the same process.

Instructor: Md. Mamun Sarkar

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