MATRICES AND SYSTEM OF LINEAR EQUATIONS

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MATRICES AND SYSTEM OF LINEAR EQUATIONS

• CHAPTER 5
• DCT1043

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CONTENT

• 5.1 Definition and Types of Matrices
• 5.2 Operation on Matrices
• 5.3 Determinant of Matrices
• 5.4 Inverse Matrices
• 5.5 System of Linear Equations
• 5.51 Solve using inverse matrix
• 5.52 Solve using Cramers Rule
• 6.53 Solve using Gauss Gauss Jordan
• Elimination Method

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5.1 Definitions Types of Matrices

• CHAPTER 5
• DCT1043

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OBJECTIVES

• By the end of this topic, you should be able to
• Define a matrix and the equality of matrices
• Identify the different types of matrices such as
  row, column, square, zero, diagonal, upper
  triangular, lower triangular, identity, and
  symmetry matrices.

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A Matrix

• A Matrix A is a rectangular
  array of numbers arranged
  
  in rows and column and
  enclosed in brackets.
• The numbers in the matrix are called the
  elements of the matrix (element in the ith row
  and jth column).
• A matrix with m rows and n columns is said to be
  of order .

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Equality of Matrices

• Two matrices are equal (A B) if
• They have the same order
• Their corresponding elements are equal
• Examples

(Different order)
(not all corresponding elements are equal)
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Types of Matrices

• Row Matrix
• A matrix with order
• Column Matrix
• A matrix with order
• Square Matrix
• A matrix with the same number of rows column
• Null (Zero) Matrix , O
• A matrix where all the elements are zero
• Identity Matrix, I
• A square matrix where the elements in the main
  diagonal are all 1s the others are all zeros

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Types of Matrices

• Diagonal Matrix
• A square matrix where all its elements zeros,
  except for those in the main diagonal
• Symmetric Matrix
• A square matrix where the elements are
  symmetrical about the main diagonal
• Upper Triangular Matrix
• A square matrix where all the elements below the
  main diagonal are zeros
• Lower Triangular Matrix
• A square matrix where all the elements above the
  main diagonal are zeros

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5.2 Operation on Matrices

• CHAPTER 5
• DCT1043

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OBJECTIVES

• By the end of this topic, you should be able to
• Perform operations on matrices such as addition,
  subtraction, scalar multiplication, and
  multiplication of two matrices.
• Define the transpose of a matrix and show its
  properties

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1. Addition

• Two matrices A and B of the same order can be
  added by adding the corresponding elements of A
  and B .
• Addition of matrices
• Commutative, that is
• Associative, that is

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Example 1 (Addition)

• Given the matrices
• find
  the following
• matrices if exist.
• If show that C O C .
  
  
  

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2. Subtraction

• Two matrices A and B of the same order can be
  subtracted (A - B ) by subtracting the
  corresponding elements of B from A.

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Example 2 (Subtraction)

• Given the matrices
• Find the
  following
• matrices if exist.
• If show that D B A
  , find
• the values of p, q and r.
  
  

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3. Scalar Multiplication

• For a matrix A and real number k, the matrix kA
  is obtained by multiplying each element in A by k.

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Example 3 (Scalar multiplication)

• Given the matrices
• find the following matrices if exist.

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4. Multiplication

• The product AB is defined only if the number of
  columns of A is equal to the number of rows of B.
• In general,
• Multiplication is done by
• Taking a row from the 1st matrix a column from
  the 2nd matrix.
• Multiplying the corresponding elements from the
  row column.
• Adding the products.

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Example 4 (Multiplication)

• Given the matrices
• find the following matrices if exist.

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Example 4 (Multiplication)

• Find the following matrices if exist

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4. Multiplication

• Multiplication of matrices is
• Associative, that is
• Distributive, that is
• In general

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Example 5 (Multiplication)

• Given the matrices
• find the following matrices if exist.
• Is AB BA ?

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5. Transpose of a matrix

• The transpose of a matrix is the
  
• matrix obtained by interchanging the rows
  
• column.
• The transpose of A is denoted by .
• Example,
• Properties

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Example 6 (Transpose)

• Given the matrices
• find the following matrices if exist.

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5.3 Determinant of Matrices

• CHAPTER 5
• DCT1043

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5.3 Determinant of Matrices

• By the end of this topic, you should be able to
• Define the determinant, minor, cofactor and
  adjoint of a square matrix
• Discuss the properties of determinants

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Determinant of Matrices
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Example 7 (Determinant)

• Given the matrices
• find its determinant if exist.
• Then show that

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Determinant of Matrices

• Use Special Formula

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Example 8 (Determinant)

• Given the matrices
• find its determinant if exist.

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Minor of an element
Example
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Cofactor of an element
Example
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Matrix of Cofactors Adjoint matrix

• 2. The transpose of the matrix of cofactors of A
  is the adjoint matrix, denoted by Adj A

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Example 9 (Minor and cofactor)

• Given the matrices
• find all its minor and cofactor. Then write down
  the adjoint matrix.

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Determinant of Matrices

• The determinant of a matrix is the sum of
  the products of the i row (or j column) elements
  with their corresponding cofactors

TIPS Choose any 1 row or column with more zero
elements.
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Example 10 (determinant)

• Given the matrices
• Use cofactor to find its determinant.

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5.4 Inverse Matrices

• CHAPTER 5
• DCT1043

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OBJECTIVES

• By the end of this topic, you should be able to
• Define the inverse of a matrix its properties
• Apply the elementary row operation to obtain the
  inverse of matrices
• Find the inverse matrix using the adjoint matrix

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Matrix inverse

• A square matrix A is invertible or nonsingular if
  there exist a square matrix B, called an inverse
  of A, such that
• B is called an inverse of A and
• A is called an inverse B
• An invertible matrix A has only one inverse (The
  inverse is unique) is denoted by .
  So,
• When does not exist and A is
  called noninvertible or singular matrix

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Properties of Inverse

• If A and B are nonsingular matrices (invertible),
  
• The inverse of an invertible matrix is also is
  invertible. So,
• Any nonzero scalar product of an invertible
  matrix is invertible.

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Inverse of a Matrix
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Example 11 (Inverse)

• Given the matrices
• find its inverse if exist.

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Example 12 (Inverse)

• Given the matrices
• If AC I, find the value of k .
• If B 2 A, find the value of h .
• Find the inverse of .

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Find Inverse Matrix by Elementary Row Operation

• To find , if it exist, do the following
• Find the reduced row echelon form (by elementary
  row operation) of the matrix AI, say BC
• If B has a zero row, STOP. So, A is
  noninvertible. Otherwise, go to Step 3.
• The reduced matrix is now in the form I .
  Read the inverse .

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Reduced Row Echelon Form Matrix

• Consider the following conditions on a matrix
• All zero rows are at the bottom of the matrix (at
  least one row is nonzero)
• The leading entry of each nonzero row after the
  first occurs to the right of the leading entry of
  the previous row.
• The leading entry is any nonzero row is 1.
• All entries in the column above and below a
  leading 1 are zero
• If a matrix satisfies the 1st two conditions, it
  is in row echelon form
• If a matrix satisfies the all four conditions, it
  is in reduced row echelon form

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Example of RE and RRE
RE
RE, RRE
RE, RRE
Fail condition 4
Fail condition 1
RE, RRE
RE
Fail condition 3
Fail condition 2
RE, RRE
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Elementary Row Operation

• The elementary row operation of a matrix consist
  of the following
• Elimination Adding a constant multiple of one
  row to another
• Scaling Multiplying a row by a nonzero constant
• Interchange Interchanging two row

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Example 13 (Inverse)

• Given the matrices
• Use Elementary Row Operation to find its
  inverse.

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Find Inverse Matrix by Adjoint Matrix
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Example 14 (Inverse)

• Given the matrices
• Use adjoint matrix to find its inverse.

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5.5 System of Linear Equations

• CHAPTER 5
• DCT1043

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OBJECTIVES

• By the end of this topic, you should be able to
• Discuss system of linear equations and the types
  of solution namely unique, inconsistent and
  infinite solutions.
• Write a system of linear equations in matrix form
• Solve a system of linear equation by using
  inverse matrix, Cramers Rule, and Gauss
  Gauss-Jordan Elimination Method.

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What is system?

• is an assemblage of entity/objects, real or
  abstract, comprising a whole with each and every
  component/ element interacting or related to
  another one.
• Solar system, blood system, computer system,
  ext..

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System of Linear Equations
The system of linear equations
where
Can be written in matrix form as
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Augmented Matrix
For the system of linear equations
where
The augmented matrix is given by,
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Types of solution
m Number of Row n Number of
Column Unique only 1 solution (the system is
consistent) Infinite many solution (the system
is consistent) None No solution (the system is
not consistent)
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5.51 Solve using Inverse Matrix

• Only for Square matrix
• The formula given by

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Example 15 (Solve using Inverse Matrix)
Solve each of the following system of equality by
Inverse Matrix
A
B
C
D
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5.52 Solve Using Cramers Rule

• Only for Square matrix
• The formula given by

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Example 16 (Solve Using Cramers Rule)
Solve each of the following system of equality by
Cramers Rule
A
B
C
D
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5.53 Solve Using Gauss Gauss-Jordan
Elimination Method

• For any matrix
• Gauss Elimination Method
• Reduce the augmented matrix Ab into row
  echelon form
• Starting with the last nonzero row, use
  back-substitution to find X
• Gauss-Jordan Elimination Method
• Reduce the augmented matrix Ab into reduced
  row echelon form IX

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Example 17 (Solve Using Gauss Gauss-Jordan
Elimination Method)
Solve each of the following system of equality by
Gauss Gauss-Jordan Elimination Method
A
B
C
D
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Example 18 (Solve system of equation )

• Use inverse matrix, Cramers Rule, and Gauss
  Gauss-Jordan Elimination Method to solve the
  following system of equation. Compare your answer.

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THANK YOU

• CHAPTER 5
• DCT1043