Matriks Dan Ruang Vektor : Sistem Persamaan Liniear Dan Representasi Matriks

Matriks dan Ruang Vektor : Sistem Persamaan Liniear dan Representasi Matriks

Introduction to Systems of Linear Equations

Linear Equation

Definition of Linear Equation is a linear equation in the 𝑛 variables 𝑥1,𝑥2,…,𝑥𝑛 to be one that can be expressed in the form

𝑎1 𝑥1 + 𝑎2 𝑥2 + ⋯+ 𝑎𝑛 𝑥𝑛 = 𝑏

Where 𝑎1,𝑎2,…,𝑎𝑛 and 𝑏 are constants (the 𝑎′s are not all zero). There is a special case, which 𝑏 = 0 called homogeneous linear equation. A linear equation with two variables called a line. However, for three variables called a plane.

For Example :

As you can see the linear equation has exactly single type variable ( which not squared, root, nor higher ) for each variable that used in linear equation. Different from linear equation, non-linear equation, however, it has more than one type of variable with consists squared, root, and trigonometry sequences. Those sequences wasn't classified as linear equation and cannot be expressed as matrix form.

System of Linear Equation

Finite set of linear equation is system of linear equations (a linear system )

For example :

A general linear system of 𝑚 equations in the 𝑛 unknowns 𝑥1,𝑥2,…,𝑥𝑛 can be written as

Definition of Solution in Linear Algebra Sequence

Solution is a sequence of 𝑛 numbers 𝑠1,𝑠2,…,𝑠𝑛 for which the substitution 𝑥1 = 𝑠1,𝑥2 = 𝑠2,…,𝑥𝑛 = 𝑠𝑛 makes each equation a true statement

Exercise 1

Proof those equation that already has the solution with right answer

The three possible types of solution

1. No solution ( linear system is inconsistent ),

ex
𝑥 + 𝑦 = 4
3𝑥 + 3𝑦 = 6

2. Exactly one solution ( linear system is consistent )

ex
𝑥 − 𝑦 = 1
2𝑥 + 𝑦 = 6

3. Infinitely many solution ( linear system is consistent )

ex
4𝑥 − 2𝑦 = 1
16𝑥 − 8𝑦 = 4

Augmented Matrices

We can abbreviate the system by writing only the rectangular array of numbers

The matrix is called the augmented matrix

Elementary Row Operations

Basic method for solving linear system: perform algebraic operations that do not alter the solution set

• The algebraic operations: 
• Multiply an equation through by a nonzero constant 
• Interchange two equations 
• Add a constant times one equation to another 

• These three operations correspond to elementary row operations on a matrix: 
• Multiply a row through by a nonzero constant 
• Interchange two rows
• Add a constant times one row to another

Using Elementary Row Operations

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Matriks Dan Ruang Vektor : Gaussian Elimination Dan Metode Operasi Baris Elementer Tereduksi ( Reduced Row Echelon Form ) Dalam Matriks

Matriks dan Ruang Vektor : Gaussian Elimination dan Metode Operasi Baris Elementer Tereduksi ( Reduced Row Echelon Form ) dalam Matriks

Metode Operasi Baris Elementer Tereduksi ( Reduced Row Echelon Form )

Properties :

1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading 1
2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix
3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row
4. Each column that contains a leading 1 has zeros everywhere else in that column

A matrix that has the first three properties is said to be in row echelon form

Example 1

Row Echelon Form :

Reduced Row Echelon Form :

Exercise 1

Determine whether the matrix is in row echelon form, reduced row echelon form, or neither :

Gauss-Jordan Elimination

The procedure for reducing matrix to row echelon form is called Gauss elimination. The procedure for reducing matrix to reduced row echelon form is called Gauss-Jordan elimination. Basically, Gauss-Jordan Elimination is just Gauss Elimination but instead changing all element into reduced row echelon form or MEBT ( Matriks Elementary Baris Tereduksi )--look other post to re-learn about MEBT Form--RRE form.

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Exercise 2

Solve the linear system by Gaussian elimination :

Solve the linear system in 1,2, and 3 by Gauss-Jordan elimination

Homogeneous System of Linear Equations

The system has the form as shown below :

Every homogeneous system of linear equations is consistent (trivial solution). If there are other solution, that's called nontrivial solutions

Use Gauss-Jordan elimination to solve the homogeneous linear system--which has 0 as a result of each equation :

The Zero Theorem

"

If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n-r free variables

"

Exercise 3

Use Gaussian elimination or Gauss-Jordan elimination to solve the homogeneous linear system :

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Matriks Dan Ruang Vektor : Dasar-Dasar Matriks Dan Operasi Matriks

Matriks dan Ruang Vektor : Dasar-dasar Matriks dan Operasi Matriks

Matrix

Definition of a matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix

Example :

Size of matrix: number of rows × number of columns. Example : for the matrices above: 2×2, 1×3, 3×3, 1×1, respectively

Matrix with only one row is called row matrix. Thus, matrix with only one column is called column matrix. We will use capital letters to denote matrices and lowercase letter to denote numerical quantities

Example :

Entry that occurs in row i and column j of matrix A will be denoted by a_ij

Square Matrix

A matrix A with n rows and n columns is called a square matrix of order n. The shaded entries a_11,a_22,…,a_nn are said to be on the main diagonal of A

Equality of Matrices

Two matrices are defined to be equal if they have same size and their corresponding entries are equal

Example :

Exercise 1

Let

What is x and y so that A = B ?

Addition and Subtraction

If A and B are matrices of the same size :

Consider the matrices

Then

The expression A+C, B+C, A-C, and B-C are undefined

Scalar Multiples

Syntax for Scalar Multiples :

Example : For the matrices

We have

Multiplying Matrices

If A is an m×r matrix and B is an r×n matrix, then the product AB is the m×n matrix whose entries are determined as follows. To find the entry in row i and column j of AB, we can do : 

• Single out row i from the matrix A and column j from the matrix B 
• Multiply the corresponding entries from the row and column together 
• Add up the resulting products 

The product of two matrices is defined if the inside numbers are the same

Example 1

Consider the matrices

For example, the entry in row 2 and column 3 of AB :

Example 2

Consider a system of m linear equations in n unknowns :

The equations above can be written as a

The matrix A is called the coefficient matrix

Transpose Matrix

If A is any m×n matrix, then the transpose of A (A^T ): the n×m matrix that results by interchanging the rows and columns of A; 

Example :

Exercise 2

Determine A^T!

Trace

If A is a square matrix, then the trace of A (tr(A)): sum of the entries on the main diagonal of A.

Example :

Sumber 

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Slide MRV Dasar-dasar Matriks

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Matriks Dan Ruang Vektor : Inversi Matriks, Properti Dalam Matriks, Dan Bentuk Matriks

Matriks dan Ruang Vektor : Inversi Matriks, Properti dalam Matriks, dan Bentuk Matriks

Properties of Matrix Arithmetic

Assuming that the sizes of the matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid

1. A+B = B+A ( commutative law for addition )
2. A+(B+C) = (A+B)+C ( Associative law for addition )
3. A(BC) = (AB)C ( Associative law for multiplication )
4. A(B+C) = AB+AC ( left distributive law )
5. (A+B)C = AC+BC ( right distributive law )
6. a(B+C) = aB+aC
7. (a+b)C = aC+bC
8. a(bC) = (ab)C
9. a(BC) = (aB)C = B(aC)

Identity Matrix

A square matrix with 1’s on the main diagonal and zeros elsewhere is called an identity matrix

 

Example

AI = IA = A 

Inverse Matrix

A is a square matrix, and if a matrix B of the same size can be found such that AB=BA=I, then 

• A is invertible (nonsingular) 
• B is an inverse of A (A is an inverse of B) 
• If C is also an inverse of A, then B=C 

If no such matrix B can be found → A is singular. If A is invertible, the its inverse will be denoted by A^(-1)

The matrix of

is invertible if and only if ad-bc ≠ 0 and

Example 1

Determine whether the matrix is invertible. If so, find its inverse.

Some Theorem of Inverse Matrices

If A and B are invertible matrices with the same size, then AB is invertible and (AB)^(-1)=B^(-1) A^(-1)

If A is invertible and n is a nonnegative integer, then:

• A^(-1) is invertible and (A^(-1) )^(-1)=A
• A^n is invertible and (A^n )^(-1)=A^(-n)=(A^(-1) )^n
• kA is invertible for any nonzero scalar k, and (kA)^(-1)=k^(-1) A^(-1)

If the sizes of the matrices are such that the stated operations can be performed, then:

• ( A^T  )^T = A
• ( A+B )^T = A^T + B^T
• ( A-B )^T = A^T- B^T
• ( kA )^T = kA^T
• ( AB )^T = B^T A^T

Elementary Matrices and a Method for Finding A^(-1)

Some Equivalent Statements

If A is an m×n matrix, then the following statements are equivalent, that is, all true or all false

1. A is invertible
2. Ax=0 has only the trivial solution
3. The reduced row echelon form of A is I_n

Inversion Algorithm

To find the inverse of an invertible matrix A : 

1. Find a sequence of elementary row operations that reduces A to the identity
2. Perform that same sequence of operations on I_n to obtain A^(-1)

Example

Find the inverse of

Solution

Exercise 1

Determine A^(-1) (if exist) ! 

Inverse and Solution of Linear System

If A is an invertible n×n matrix, then for each n×1 matrix b, the system of equations Ax=b has exactly one solution, namely 

x=A^(-1) b

Exercise 2

Solve the linear equations using A^(-1)

Diagonal, Triangular, and Symmetric Matrices

Diagonal Matrices

A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix. 

Example

A general n×n diagonal matrix D can be written as

Example 2

If

Determine the value of A^(-1), A^5,A^(-5) !

Solution

Upper and Lower Matrices

A square matrix A=[a_ij] is upper triangular if and only if all entries to the left of the main diagonal are zero; that is, a_ij=0, if i > j

Symmetric Matrices

A square matrix A is said to be symmetric if A = A^T 

(A)_ij = (A)_ji 

Example

Sumber

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Slide MRV : Properties of Matrices and Inverse

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